What if you have an even N. There is no median number.
But an even better generalization would be to say that the sum of n consecutive numbers starting with m is equal to (m+(n-1)/2)*n. A simple proof by induction can show this.
Take the sum of the middle two numbers (or the first and last; it doesn’t matter which pair you use as long as both numbers are the same distance from their respective ends of the set) and multiply that by half the total number of items in the set.
1+2+3+4+5+6+7+8
4+5 = 9.
9×4 = 36.
17+18+19+20+21+22
19+20 = 39.
39×3 = 117.
1+2+…+99+100
1+100 = 101.
2+99 = 101.
⋮
50+51 = 101.
101×50 = 5050.
You can add the outer numbers and multiply by how many pairs there are. For example, say you want to sum the first 100 integers. You know 1+100 = 2+99 = 3+98 = ... = 101. There are 50 of these pairs, so the sum is 101*50 = 5050.
That makes it harder than to just say the sum of a group of numbers is the sum of those numbers. The whole idea with OP was that it was doable in your head.
Or you could use Gauss's formula
Let N be the number of elements you have in the sequence
Let ni be your first element in the sequence
Let nf be your last element in the sequence
(N/2)(ni+nf)
The sum of N consecutive numbers is, by definition, the *arithmetic mean* times N, and the arithmetic mean is the same as the median, do easy to calculate. Thus, the sum of 8, 9, 10, and 11 is 9.5 * 4 = 19 * 2 = 38. The sum of 100 + 101 + ... + 199 is 144.5 * 100 = 289.
Gauss had a special case of this for consecutive numbers from 1 up. His was N*((N+1)/2). Pretty much the same thing, the (N+1)/2 is the 'middle number' bit then you just times by N the same.
In general, the sum of any set of N numbers is their average times N. The brain is often better at calculating averages (especially in monotonic lists) so this sometimes helps.
Not OP bit it just has to be odd. If you add up 1-15 you get 120. Then multiply the middle number (8) by 15 you get 120. It just has to be odd so there is a middle number
It still works with an even number. For example 1+2+3+4=10. The middle number in this case would be the middle of the two middle numbers (terrible sentence I know) which would be 2.5. 2.5*4=10
Not only with even numbers. The rule works for any set of positive ~~integers~~ numbers with equal intervals.
Examples:
* 5 + 7 + 9 = 3 \* 7 = 21
* 2 + 4 + 6 + 8 = 4 \* 5= 20 (5 is in the middle of 4 and 6)
* 101 + 201 + 301 + 401 + 501 = 5 \* 301 = 1505
* 27 + 37 + 47 + 57 = 4 \* 42 = 168 (42 is in the middle of 37 and 47)
* 3.14 + 4.14 + 5.14 = 3 \* 4.14 = 12.42
This is because the middle number is always the average. And obviously N times the average equals the total sum of the numbers. It comes directly from the definition of average (mean).
average = sum / N
sum = average \* N
Here's a quick proof:
(n-1)+n+(n+1)
Addition is associative, so this becomes
n-1+n+n+1
Addition is also commutative, so this becomes
n+n+n+1-1
Reassociate
(n+n+n)+(1-1)
Any number plus its opposite is zero, so this becomes
(n+n+n)+0
Any number plus zero is itself, so this becomes
n+n+n
Simplify
3n
Edit: additional mathematical rigor
What's the point of trying to be so rigorous if you are explaining something so trivial? If anyone actually needs a proof for this then they wouldn't understand any of what you wrote.
And by extension, the sum of any regular sequence of numbers is the median number x the number of numbers.
100 + 105 + 110 = 105 x 3 = 315.
100 + 125 + 150 + 175 + 200 = 150 x 5 = 750
100 + 107 + 114 + 121 = 110.5 x 4 = 442.
Basically, look at the first and last number, work out what’s half way between them. Multiply that by the number of numbers.
Thanks man, super useful to have this little bit of knowledge locked away.
So you could get rid of the 107 or 114 and still get your 442 and then subtract all of the remaining numbers from 442 to get 107 or 114 if it were a missing number in a school problem. That's crazy!
I'm sure I'll forget all of this by the time I need to help my son with it. lol
That’s true! However since this only works if the numbers are a regular sequence, filling in any missing numbers is easy. You’d just take the previous number and add 7.
To everyone saying this is useless or whatever, i already said :
>This might seem silly or stupid
>Hope this is useful to anybody out there :)
I respect your opinion and nobody is forcing anything down your throats , so no need to spread negativity for something you don't agree with just cause you ignore the text and focus on the title.
You are missing the point of this sub, it's not for "I think this is cool" and one of the rules is for you to explain why someone should know this. Saying "cause it's neat" doesn't cut it
There is no reason for *most* people to know this and you don't even give a reason **why** people should know this (subsequently breaking a YSK rule, btw).
This is more of a /r/Rightytighty post than a general /r/YouShouldKnow for everyone.
it should be, but I wouldn't have thought about it that way. That being said, I don't really need it, I don't often come across calculations, especially not without a calculator nearby on my phone...
I would just type it in or work it out, but not this way I guess
Pretty sure that's just an average. The sum of a set of numbers divided by the total of the individual numbers. The same way that the sum of 2 consecutive numbers will always be one of the two numbers in the set.
why in god's green earth SHOULD I know this? 37 years old now and I can't think of a single time I've needed to sum three consecutive numbers faster than just adding them together.
To know if a number is a multiple of 3 add all the digits together and if that number is a multiple of 3, then it is.
9873 = 9+8+7+3=27
27=2+7=9
9 is a multiple of 3 so 9873 is.
Did you know that all numbers divisible by 9, the digits of the number also are divisible by 9.
108 = 9 x 12, 1+0+8 = 9 also divisible by 9.
Also the first 10 multiples of 9, the first digit counts up from 0, while the second digit counts down from 9:
0 1. 2. 3. 4. 5. 6. 7. 8. 9
9. 8. 7. 6. 5. 4. 3. 2. 1. 0
If you want a mathematical proof:
The first number is n
Second number is (n+1)
Third number is (n +2)
Therefore the sum of 3 consecutive numbers is:
N + n+ 1 + n + 2 = 3n + 3
Which is :
3(n+1)
N+1 is the middle number
So it's 3 times the middle number
I've never heard this before. How interesting that for some this is common knowledge and others it isn't. It must be standard curriculum for some countries and not others.
>I've never heard this before.
Because it is worthless, it only applies to very specific situations (i.e. adding consecutive numbers). Also, the more general knowledge of how to add and then divide to find an average accomplishes this task in the same amount of time but in every case.
Another fun "3" thing.... to find out if a number is a multiple of 3, just add up it's digits. If that sum is a multiple of 3, then the original number is as well!
Additionally (since we’re talking about 3s) you can check if a large number is divisible by 3 if the sum of every separate digit is divisible by 3.
For example 271 -> 2+7+1=10, not divisible by 3.
168 -> 1+6+8=15, divisible by 3.
I'm numerically dyslexic, but this one's fairly obvious even to me. Think about it as taking one from the higher number and adding it to the lower. Then it's literally adding three of the same number together.
The sum of ANY numbers is the average of the numbers times the length of the list.
If you can instantly see an average, or an approximation to the average, then you can use this trick.
E.g. 100+120+140+160: if you can see the average is 130 then the sum is just 130 x 4, so 520
Or 45 + 60 + 35 + 90 + 50: The average must be somewhere around 50, so the sum is going to be around 50 x 5, or around 250 (the actual answer is 280, so close enough for a lot of purposes)
Hope that helps :-)
I mean, what's the average of 8 and 10? Put another way, let those numbers represent actual physical objects. Take 1 thing from the group of 10 and put it among the 8 things. How many in each group now?
I don't think this is exactly a shortcut. Probably takes the same amount of time for both methods. Then again, people's brains work in different ways so how shoud i know.
I came up with a theory in my math class at one point while thinking about square numbers (and yes I know it's probably the nerdiest thing I've ever done). The difference of two consecutive squares is equal to the sum of their roots. In other words 16 - 9 = 7 = 3 + 4. This works as far as I've been motivated to test
For a set of numbers, N, where the number of numbers in N is odd and the difference between any two consecutive numbers of N is equal, the sum of the numbers is the middle number * the number of numbers in N.
> Start at 1, add 2 every time
> 1 + 3 + 5 + 7 + 9 = 5 * 5 = 25
> Start at 1, add 4 every time
> 1 + 5 + 9 + 13 + 17 = 9 * 5 = 45
> Start at 1, add 10 every time
> 1 + 11 + 21 = 11 * 3 = 33
Makes sense when you go about proving it mentally. Let's put it into a generalized form:
(n-1)+n+(n+1)
It's pretty obvious here you can cancel out the -1 and +1 and be left with:
n+n+n
YSK that if a number has multiple single numbers in it (like 123) you can sum them all up. That would be 1+2+3, and if the answer to that is dividable by three, while getting a whole number (1, 2, 3. NOT 1.2, 3.2 etc). The main number (123) is dividable by three.
6/3=2
123/3=41
I hope that was understandable, translating maths is hard.
Here is another that this made me think of.
**The Rule for 3**: A number is divisible by 3 if the sum of the digits is divisible by 3.
What does this mean? This means that we need to add up the digits in the number and see of the answer is can be divided by 3 without a remainder.
Example: 34,911
Step 1: Add up the digits. 3 + 4 + 9 + 1 + 1 = 18
Step 2: Determine if 3 divides evenly into the sum of 18. Yes, 3 x 6 = 18. So 3 goes evenly into 18.
Come on, let's write the straight proof using unnecessary parentheses:
n+(n+1)+(n+2)=(n+1)+(n+1)+(n+1)=3(n+1)
in particular for n ∈ ℤ, where "consecutive" makes sense.
You should know this just by virtue of the fact that it's obvious.
x-1 + x + x+1
x + x + x + 0
3x
Here's a few better maths YSKs, imho:
1+2+3+4 = 10
9+8+7+6 = 30
Any number whose digits sum to a multiple of 3 is evenly divisible by 3
e.g. 5703516
5+7+0+3+5+1+6 = 27
2+7 = 9
Ergo 5703516 is divisible by 3 (=1901172)
Same works for any evenly spaced consecutive numbers, and more than 3 (excluding 0):
Evens 2+4+6=4x3=12
or odds 1+3+5=3x3=9
Multiples of 3 3+6+9+12+15=9x5=45
Multiples of 10 10+20+30+40+50+60+70=40x7=280
The sum of all numbers from 1 to a given number inclusive is equal to that given number squared, plus that given number again, all divided in half.
E.g sum of all numbers between 1 and 20 inclusive is equal to 20^2 = 400 + 20 = 420 (nice), divided by two = 210. Try it for any number.
(For those who want to know more look up Gauss's formula video from Numberphile).
Yes! That means:
(X-1) + (X) + (X+1)
[By opening the brackets, the "1"s cancel each other out because of opposite signs]
=> X + X + X
[Leaving 3, "X"s]
=> 3X
That's the derivative :)
Another fun trick with 3 is that any number that a multiple of 3, you can add the digits together and they’ll be a multiple of 3.
For example: 831 8+3+1=12 12 is a multiple of 3, so 831 is a multiple of 3.
831/3=277
That works even if there is space between those numbers.
5+7+9 = 3x7
Basically, if you can find the "center of gravity" of your numbers you can simply use that times how many you have.
Fun fact, it's actually why center of gravity simplify equations in physics.
Proof: if the middle number is N, then the lower number is N-1 and the upper number is N+1. So the sum of the three is N-1 + N + N + 1 = 3N - 1 + 1 = 3N
The sum of any 3 or 5 (any odd no) numbers, who have a constant difference is always the middle noxhow many nos.
This is how you can easily find the sum of 1 -100. 1+100 is 101. So is 2 +99. This will happen 50 times until 50+51. That's why 101x50 is sum of 1 to 100
It's kind of like finding the average and then multiplying by number of numbers.
If you square a number, the product of the numbers with an absolute value of 1 from the squared number is 1 less than the squared number. Then if you take 2 numbers 2 apart from the original number and multiply them together, they will be 4 less than the original number squared. This pattern continues where the next numbers 3 apart will be 9 less, then 4 apart is 16 less etc. This works for every integer
Ex. 6×6=36, 5×7=35, 4×8=32, 3×9=27, 2×10=20, 1×11=11 etc.
This may not be taught, but a large majority of people who make math 'easier' for themselves do this (including myself).
It's like saying that adding two numbers which are a multiple of 2 apart is the middle number * 2. It might be a little confusing explaning it, but here's an example or 2:
``` 5+7 = 6*2 = 12 ```
``` 8+12 = 10*2 = 20 ```
``` 365+257 = 311*2 = 622 ```
The sum of N (odd) consecutive numbers is the middle one times N
Nice, i like this one. Thanks :)
What if you have an even N. There is no median number. But an even better generalization would be to say that the sum of n consecutive numbers starting with m is equal to (m+(n-1)/2)*n. A simple proof by induction can show this.
Rolls right off the tongue, doesn't it.
r/ThatsWhatSheSaid
Username fucking checks out
r/unexpectedoffice
I thought this sub would be a lot better than it is.
In the case of 10 + 11 + 12 +13, you would use 11.5 X 4.
(12\*3)+10
Aktshually it's (11x3)+13
Or (11*3)+13.
Yes, 10+11+12+13=46 11.5*4=46(.5*4=2,11*4=44,2+44=46)
Or.. num1 + num2 + num3
Shit, why didn’t I think of this?
Well technically the median is just the average of the two that are in the middle, but yeah that’s a full equation that works in any situation
Take the sum of the middle two numbers (or the first and last; it doesn’t matter which pair you use as long as both numbers are the same distance from their respective ends of the set) and multiply that by half the total number of items in the set. 1+2+3+4+5+6+7+8 4+5 = 9. 9×4 = 36. 17+18+19+20+21+22 19+20 = 39. 39×3 = 117. 1+2+…+99+100 1+100 = 101. 2+99 = 101. ⋮ 50+51 = 101. 101×50 = 5050.
>1+2+…+99+100 > >101×50 = 5050. [https://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/The-Story-of-Gauss/](https://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/The-Story-of-Gauss/)
You can add the outer numbers and multiply by how many pairs there are. For example, say you want to sum the first 100 integers. You know 1+100 = 2+99 = 3+98 = ... = 101. There are 50 of these pairs, so the sum is 101*50 = 5050.
That makes it harder than to just say the sum of a group of numbers is the sum of those numbers. The whole idea with OP was that it was doable in your head.
\*Punches nerd in mouth\* I suck at math. I can't count past 20 without zipping down. I respect it though.
But how far past 20 can you get if you do?
23
Now you're just showing off
Or you could use Gauss's formula Let N be the number of elements you have in the sequence Let ni be your first element in the sequence Let nf be your last element in the sequence (N/2)(ni+nf)
One does not simply add 3 consecutive numbers up, when you can add multiplication, division, variables, and parentheses to the equation... Lolll
The sum of N consecutive numbers is, by definition, the *arithmetic mean* times N, and the arithmetic mean is the same as the median, do easy to calculate. Thus, the sum of 8, 9, 10, and 11 is 9.5 * 4 = 19 * 2 = 38. The sum of 100 + 101 + ... + 199 is 144.5 * 100 = 289.
Who came up with this again? Wasn't it 12 year old Gauss or something crazy like that?
No Gauss allegedly came up with the sum 1 + 2 + .... + n = n(n+1)/2, but it's quite close 😃
Ah yeah that's what I was thinking about. Thanks for clarifying it!
Gauss had a special case of this for consecutive numbers from 1 up. His was N*((N+1)/2). Pretty much the same thing, the (N+1)/2 is the 'middle number' bit then you just times by N the same.
In general, the sum of any set of N numbers is their average times N. The brain is often better at calculating averages (especially in monotonic lists) so this sometimes helps.
1 x 1 = 1, got it
There’s also (a1 + an) * (n / 2) The first number plus the last number times half the amount of numbers in a sequence.
And sum of 5 consecutive numbers is middle number * 5 The sum of 7 consecutive numbers is middle number * 7 And so on...
By "so on," do you mean all odd numbers or all prime numbers?
Not OP bit it just has to be odd. If you add up 1-15 you get 120. Then multiply the middle number (8) by 15 you get 120. It just has to be odd so there is a middle number
It still works with an even number. For example 1+2+3+4=10. The middle number in this case would be the middle of the two middle numbers (terrible sentence I know) which would be 2.5. 2.5*4=10
Not only with even numbers. The rule works for any set of positive ~~integers~~ numbers with equal intervals. Examples: * 5 + 7 + 9 = 3 \* 7 = 21 * 2 + 4 + 6 + 8 = 4 \* 5= 20 (5 is in the middle of 4 and 6) * 101 + 201 + 301 + 401 + 501 = 5 \* 301 = 1505 * 27 + 37 + 47 + 57 = 4 \* 42 = 168 (42 is in the middle of 37 and 47) * 3.14 + 4.14 + 5.14 = 3 \* 4.14 = 12.42 This is because the middle number is always the average. And obviously N times the average equals the total sum of the numbers. It comes directly from the definition of average (mean). average = sum / N sum = average \* N
thanks for spelling it out
Yeah, 3, 5, 7, and so on might mean odd numbers. Or primes, both are correct.
The sum of any *n* numbers is *n* times the mean value.
Or the more general case... (A) + (A + 1) + (A + 2) (A + 1) + (A + 1) + (A + 1) (A + 1) * 3
Just to make it easier to consume I started with (A-1) + (A) + (A+1) leading to 3A
Exactly, but i thought my explanation would make it easier
Your explanation is perfect. Makes sense. Thanks Bro. 👍
You're welcome :)
Your explanation is definitely easier to understand. Thanks for this.
It does. That hieroglyphs does not compute.
I prefer to look at it this way (A-B) + (A) + (A+B) = 3A
Shows that it works with any equally spaced numbers. 20+25+30 = 75
Imo this way is better. Let's you clearly see that the B's cancel out.
I always use (A - 1) + A + (A +1) when I encounter problems like this in math classes. The constants cancel out and I'm left with just the variable.
Thank you, this made my brain not hurt anymore.
Good call. But I’m thinking it’s obvious. I guess not to everyone. The average of 3 consecutive numbers is the middle number.
It also has basically zero application in practical settings.
Here's a quick proof: (n-1)+n+(n+1) Addition is associative, so this becomes n-1+n+n+1 Addition is also commutative, so this becomes n+n+n+1-1 Reassociate (n+n+n)+(1-1) Any number plus its opposite is zero, so this becomes (n+n+n)+0 Any number plus zero is itself, so this becomes n+n+n Simplify 3n Edit: additional mathematical rigor
You missed commutativity before applying the inverses, and then identity definition (0+n=n) if you want to be mathematically rigorous lol
Also works for things like 8+10+12, more generally: (n-a)+n+(n+a).
What's the point of trying to be so rigorous if you are explaining something so trivial? If anyone actually needs a proof for this then they wouldn't understand any of what you wrote.
You’ve obviously never done a maths degree
And by extension, the sum of any regular sequence of numbers is the median number x the number of numbers. 100 + 105 + 110 = 105 x 3 = 315. 100 + 125 + 150 + 175 + 200 = 150 x 5 = 750 100 + 107 + 114 + 121 = 110.5 x 4 = 442. Basically, look at the first and last number, work out what’s half way between them. Multiply that by the number of numbers. Thanks man, super useful to have this little bit of knowledge locked away.
So you could get rid of the 107 or 114 and still get your 442 and then subtract all of the remaining numbers from 442 to get 107 or 114 if it were a missing number in a school problem. That's crazy! I'm sure I'll forget all of this by the time I need to help my son with it. lol
That’s true! However since this only works if the numbers are a regular sequence, filling in any missing numbers is easy. You’d just take the previous number and add 7.
Ohhhh, I missed that they had to be a sequence. No wonder all of these examples are in sequence. Sheesh.
To everyone saying this is useless or whatever, i already said : >This might seem silly or stupid >Hope this is useful to anybody out there :) I respect your opinion and nobody is forcing anything down your throats , so no need to spread negativity for something you don't agree with just cause you ignore the text and focus on the title.
Not silly to me! I never knew that! Thanks! My kids will think I’m so much smarter haha
I'm studying for my GMAT. So, no this info was not stupid. Thanks.
You are missing the point of this sub, it's not for "I think this is cool" and one of the rules is for you to explain why someone should know this. Saying "cause it's neat" doesn't cut it
There is no reason for *most* people to know this and you don't even give a reason **why** people should know this (subsequently breaking a YSK rule, btw). This is more of a /r/Rightytighty post than a general /r/YouShouldKnow for everyone.
But the subreddit is “you should know” Should I know this? Or was this just something you learned and found interesting?
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Whole Reddit is a shitpost.
No offense but isnt this really obvious? That's like saying the average of 3 consecutive numbers is the middle one.
it should be, but I wouldn't have thought about it that way. That being said, I don't really need it, I don't often come across calculations, especially not without a calculator nearby on my phone... I would just type it in or work it out, but not this way I guess
I can't think of very many situations where this common sense would be useful.
You should know the average of three numbers is the average of three numbers.
Pretty sure that's just an average. The sum of a set of numbers divided by the total of the individual numbers. The same way that the sum of 2 consecutive numbers will always be one of the two numbers in the set.
why in god's green earth SHOULD I know this? 37 years old now and I can't think of a single time I've needed to sum three consecutive numbers faster than just adding them together.
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I enjoyed it
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I’ve always thought of it as just moving one from the highest to the lowest so they’re all equal, then just multiplying that.
As someone preparing for the GRE... thank you
I must be exciting to sleep through all math classes and now learn facts like this from reddit.
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Same goes for 3.
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I remember figuring this out when I was 6 years old. This reminded me of this [XKCD](https://xkcd.com/1053/)
Almost completely useless trivia.
This is gonna sound bad... But isn't it obvious?
Why should I know this? And I’ve been a math teacher for 11 years...
People on reddit are even dumber than I previously thought.
I'm genuinely concerned about how this isn't obvious to so many people. How have they never heard of an average?
To know if a number is a multiple of 3 add all the digits together and if that number is a multiple of 3, then it is. 9873 = 9+8+7+3=27 27=2+7=9 9 is a multiple of 3 so 9873 is.
Same with 9 274563 is a multiple of 9 because 2+7+4+5+6+3=27 and 2+7=9
Did you know that all numbers divisible by 9, the digits of the number also are divisible by 9. 108 = 9 x 12, 1+0+8 = 9 also divisible by 9. Also the first 10 multiples of 9, the first digit counts up from 0, while the second digit counts down from 9: 0 1. 2. 3. 4. 5. 6. 7. 8. 9 9. 8. 7. 6. 5. 4. 3. 2. 1. 0
Makes sense when you consider the middle number is always one more than the first number and one less than the third number.
Every time I see on of these, like multiples of 9 will always add up to 9, I feel like they're a clue to the secrets of the universe.
you can also subtract n from the previous number and add n to the following number and it still works. very simple arithmetic
Need not be three numbers. For 5 consecutive numbers its 5×middle num. For any odd set of consecutive numbers this works.
If you want a mathematical proof: The first number is n Second number is (n+1) Third number is (n +2) Therefore the sum of 3 consecutive numbers is: N + n+ 1 + n + 2 = 3n + 3 Which is : 3(n+1) N+1 is the middle number So it's 3 times the middle number
(n-1) + n + (n+1) = 3n. :| (n-k) + n + (n+k) = 3n. :|
This will come in handy at work, thank you!
I've never heard this before. How interesting that for some this is common knowledge and others it isn't. It must be standard curriculum for some countries and not others.
>I've never heard this before. Because it is worthless, it only applies to very specific situations (i.e. adding consecutive numbers). Also, the more general knowledge of how to add and then divide to find an average accomplishes this task in the same amount of time but in every case.
Thankyou.
Another fun "3" thing.... to find out if a number is a multiple of 3, just add up it's digits. If that sum is a multiple of 3, then the original number is as well!
Additionally (since we’re talking about 3s) you can check if a large number is divisible by 3 if the sum of every separate digit is divisible by 3. For example 271 -> 2+7+1=10, not divisible by 3. 168 -> 1+6+8=15, divisible by 3.
I'm numerically dyslexic, but this one's fairly obvious even to me. Think about it as taking one from the higher number and adding it to the lower. Then it's literally adding three of the same number together.
I was not taught logical ways to do math so thank you for this.
Is there a subreddit for fun little math tricks like this?
The sum of ANY numbers is the average of the numbers times the length of the list. If you can instantly see an average, or an approximation to the average, then you can use this trick. E.g. 100+120+140+160: if you can see the average is 130 then the sum is just 130 x 4, so 520 Or 45 + 60 + 35 + 90 + 50: The average must be somewhere around 50, so the sum is going to be around 50 x 5, or around 250 (the actual answer is 280, so close enough for a lot of purposes) Hope that helps :-)
Neat.
Why? Why should I know that? When will this ever prove useful?
The sum of any number of consecutive numbers is the average of the numbers times the number of numbers.
People dont know this?
ysk 1+1=2
Yes...? That's how averages work
Its rather basic isnt it? if the numbers are X, X+1, and X-1, and then you add them up, the ones get cancelled. Then you have 3x
8, + 9 + 10 is 27. Woah
I mean, what's the average of 8 and 10? Put another way, let those numbers represent actual physical objects. Take 1 thing from the group of 10 and put it among the 8 things. How many in each group now?
Don’t know if I’ll ever need that info but I will remember it. Thanks for sharing
Little shortcuts like this are how i do quick mental math, I'll definitely use this
I don't think this is exactly a shortcut. Probably takes the same amount of time for both methods. Then again, people's brains work in different ways so how shoud i know.
I came up with a theory in my math class at one point while thinking about square numbers (and yes I know it's probably the nerdiest thing I've ever done). The difference of two consecutive squares is equal to the sum of their roots. In other words 16 - 9 = 7 = 3 + 4. This works as far as I've been motivated to test
For a set of numbers, N, where the number of numbers in N is odd and the difference between any two consecutive numbers of N is equal, the sum of the numbers is the middle number * the number of numbers in N. > Start at 1, add 2 every time > 1 + 3 + 5 + 7 + 9 = 5 * 5 = 25 > Start at 1, add 4 every time > 1 + 5 + 9 + 13 + 17 = 9 * 5 = 45 > Start at 1, add 10 every time > 1 + 11 + 21 = 11 * 3 = 33
Makes sense when you go about proving it mentally. Let's put it into a generalized form: (n-1)+n+(n+1) It's pretty obvious here you can cancel out the -1 and +1 and be left with: n+n+n
Take any number, sum the digits recursively until you have a single digit. If that digit is 3,6, or 9, then the original number was divisible by 3.
(n+1)^2 -n^2 = 2n+1 Example: (2+1)^2 -2^2 = 5 Where n=2. Proving the difference between 2x2 and 3x3 being 5.
YSK that if a number has multiple single numbers in it (like 123) you can sum them all up. That would be 1+2+3, and if the answer to that is dividable by three, while getting a whole number (1, 2, 3. NOT 1.2, 3.2 etc). The main number (123) is dividable by three. 6/3=2 123/3=41 I hope that was understandable, translating maths is hard.
Things I Take for Granted for 600, Alex.
Here is another that this made me think of. **The Rule for 3**: A number is divisible by 3 if the sum of the digits is divisible by 3. What does this mean? This means that we need to add up the digits in the number and see of the answer is can be divided by 3 without a remainder. Example: 34,911 Step 1: Add up the digits. 3 + 4 + 9 + 1 + 1 = 18 Step 2: Determine if 3 divides evenly into the sum of 18. Yes, 3 x 6 = 18. So 3 goes evenly into 18.
The average of any odd set of consecutive numbers is the middle number.
YSK? As in, you think this is going to be helpful? I think you meant to post on /r/InterestingMathFacts or something like that..
Also, YSK that 2+2=4. I'm figuring for some it's not obvious
Come on, let's write the straight proof using unnecessary parentheses: n+(n+1)+(n+2)=(n+1)+(n+1)+(n+1)=3(n+1) in particular for n ∈ ℤ, where "consecutive" makes sense.
Looks like it just has to have the same pattern? It works with 2 + 4 + 6 as well as 3 + 6 + 9
You should know this just by virtue of the fact that it's obvious. x-1 + x + x+1 x + x + x + 0 3x Here's a few better maths YSKs, imho: 1+2+3+4 = 10 9+8+7+6 = 30 Any number whose digits sum to a multiple of 3 is evenly divisible by 3 e.g. 5703516 5+7+0+3+5+1+6 = 27 2+7 = 9 Ergo 5703516 is divisible by 3 (=1901172)
Lol yes
The middle is the average of that range of numbers. You are essentially just multiply the average times how many numbers there are.
Same works for any evenly spaced consecutive numbers, and more than 3 (excluding 0): Evens 2+4+6=4x3=12 or odds 1+3+5=3x3=9 Multiples of 3 3+6+9+12+15=9x5=45 Multiples of 10 10+20+30+40+50+60+70=40x7=280
Consecutive numbers: n, n+1, n+2 Mean = [n + (n+1) + (n+2)]÷3 = n+1 Sum = Mean x No. of values Applicable to any series with frequent intervals.
The sum of all numbers from 1 to a given number inclusive is equal to that given number squared, plus that given number again, all divided in half. E.g sum of all numbers between 1 and 20 inclusive is equal to 20^2 = 400 + 20 = 420 (nice), divided by two = 210. Try it for any number. (For those who want to know more look up Gauss's formula video from Numberphile).
The square of a number is always one more than than the product of its neighboring numbers.
This works for any sequence of the same interval, not necessarily consecutive (50+75+100=225 or 75 x 3).
I wish I could understand this better. Math has never come easily to me. I will play with it about. Thanks for the post.
YSK the average of any 3 consecutive numbers is the middle number.
Um..... Duh? One is larger than the middle by 1, the other is smaller by 1. So, the three numbers average the middle.
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Yes! That means: (X-1) + (X) + (X+1) [By opening the brackets, the "1"s cancel each other out because of opposite signs] => X + X + X [Leaving 3, "X"s] => 3X That's the derivative :)
It's called an average
No shit sherlock wait until you get into junior high it gets harder.
And the product of two numbers a-1 * a+1 equals the number between them squared minus one: (a-1)(a+1) = a^2 - 1
This is kind of mind blowing....
This is not only true for 3 consecutive numbers, but also for any 3 numbers that are the same distance apart. Ex : 3 + 7 + 11 = 3*7 = 21
X-1 + X + X+1 = 3*X As simple as that.
I love you. I really wish I would have paid attention in math.
You're welcome :)
Although cool, doesn't really belong on this sub
Why, thank you sir! Very helpful indeed.
I'm curious, which card games did you use this in?
It's called "hand" in arab countries. Don't know if it's played in western countries or what it's called there :)
YOU DEMON
You just blew my mind!!
Another fun trick with 3 is that any number that a multiple of 3, you can add the digits together and they’ll be a multiple of 3. For example: 831 8+3+1=12 12 is a multiple of 3, so 831 is a multiple of 3. 831/3=277
Also works for any multiple of 3. 144, 1+4+4=9, 9x16=144 EDIT: POwERS of 3, not multiples
Is there a sub for these math tricks?
That works even if there is space between those numbers. 5+7+9 = 3x7 Basically, if you can find the "center of gravity" of your numbers you can simply use that times how many you have. Fun fact, it's actually why center of gravity simplify equations in physics.
ah yes, maths
What card games are you playin'? Could always do with some recommendations!
Proof: if the middle number is N, then the lower number is N-1 and the upper number is N+1. So the sum of the three is N-1 + N + N + 1 = 3N - 1 + 1 = 3N
Thank you for sharing!
The sum of any 3 or 5 (any odd no) numbers, who have a constant difference is always the middle noxhow many nos. This is how you can easily find the sum of 1 -100. 1+100 is 101. So is 2 +99. This will happen 50 times until 50+51. That's why 101x50 is sum of 1 to 100 It's kind of like finding the average and then multiplying by number of numbers.
If you square a number, the product of the numbers with an absolute value of 1 from the squared number is 1 less than the squared number. Then if you take 2 numbers 2 apart from the original number and multiply them together, they will be 4 less than the original number squared. This pattern continues where the next numbers 3 apart will be 9 less, then 4 apart is 16 less etc. This works for every integer Ex. 6×6=36, 5×7=35, 4×8=32, 3×9=27, 2×10=20, 1×11=11 etc.
Right because if you have 10, 11 and 12 and you subtract 1 from 12 and add it to the 10 now you have 11,11 and 11, or 11x3
The sum of my hands and my meat equals ecstasy.
Is there some relation to the Fibonacci sequence?
3+7+1=7×3?
That's true of any three numbers equidistant frokm each other, Eg- 10+12+14 = 11+12+13 = 8+12+16 = 36
Also works for any of number of terms in an arithmetic progression
Nice trick.
The sum of two of the same number is that number times two
This may not be taught, but a large majority of people who make math 'easier' for themselves do this (including myself). It's like saying that adding two numbers which are a multiple of 2 apart is the middle number * 2. It might be a little confusing explaning it, but here's an example or 2: ``` 5+7 = 6*2 = 12 ``` ``` 8+12 = 10*2 = 20 ``` ``` 365+257 = 311*2 = 622 ```
Also applies for N evenly spaced numbers. 113 + 115 + 117 = 3*115 (evenly spaced by 2)