what is every number from 1 to 500 added together equal?

Question

Like 1+2+3+4=5+6+7+8+9+10?

Answer 1

To sum the numbers:
1 + 2 + 3 + ... + n,
the formula is n(n+1)/2

In your case, n = 500
500(500+1)/2
= 500(501)/ 2
= 125,250

Answer 2

The sum of any series of number from 1 to a number n, is n*[n+1]/2

So your answer is 500*501/2 which is 125 250.

If you are up to it, I can prove this for you.
First of all the sum of 1 is 1. It is the only number. But 1 is also n*[n+1]/2 or 2*1/2 =1.

So the first number in the series fits. Now let's call 1 "n-1" nand call 2 "n", which fitting our formula the sum of all numbers from 1 to 1 far is (n-1)n/2 or (2-1)*2/2 or 1. Not very useful, but now we have a general case. What happens when we add the next number in the series? So the next number is "n". (n-1)n/2 + n. Combining the fractions we get [(n-1)*n + 2n]/2. Now multiplying (n-1)*n gives us n^2 - n. Adding the 2n, we have n^2 +1 or n(n+1), and the fraction comes out to (n)(n+1)/2. Sincce this is a general case, and we did not per se specify "n" is 2, our operation will work for any n we choose.

I realise that was a little hard to understand, but trust me it works out. Another way you can think about it for even numbers at least is if you add say 500 and 1, you get 501. Same if you add 499 and 2. And you do this all the way to 250+251 = 501, so you can make 250 or 500/2 pairs of 501s, so multiplying (500/2)*501 will give you the sum from 1 to 500. For odd numbers it's more complicated, but if you think about it those will work out too.

Answer 3

MsMath's answer is correct, but what if you don't remember the formula? Can you figure it out on your own, maybe on a test?

Here's one technique - make three columns - in the first, simply list the numbers 1, 2, 3... in the second, list the sums. In the third, try to come up with a relationship in terms of n:


1_________1_________n
2_________3_________n + 1
3_________6_________2n
4________10_________2n+2 = 2(n+1)
5________15_________3n
6________21_________3n+3 = 3(n+1)

If you look at the third column, every other one is clearly following the pattern (n/2)*(n+1). A little checking and you'll see the others follow the same pattern. Once you've established a pattern for the first few numbers, you know the pattern applies for all numbers (if you haven't learned this powerful technique for proofs, you will soon). So once you know (n/2)*(n+1) is the sum of all numbers 1 through n, it's trivial to find the solution for n=500.

PS - the grouping technique is very clever. If you can see that, that way is much simpler. I usually have to resort to plodding through the long way, unfortunately.

Answer 4

The answer is n/2(2a+(n-1)d) where a is the starting number, n is the last number and d is the common difference. Here a=1, n=500, d=1 and so the sum = 250(2+499)=250x501=125250

Answer 5

To sum the numbers:
1 + 2 + 3 + ... + n,
the formula is n(n+1)/2

In your case, n = 500
500(500+1)/2
= 500(501)/ 2
= 125,250

Answer 6

Think about it like this...
I want
1+2+...+499+500=S for sum. You want to solve for S.

Write it this way
S=1+2+...+499+500
S=500+499+...+2+1

Add the two rows together and we get
2S=501+501+...+501+501
because each term adds up to 501.

Furthermore, we know that there are 500 terms so
2S=501+501+...+501+501
2S=(501)(500)

Now, it is a lot easier to solve for S.
2S=(501)(500)=250500
S=125520

So, if I add all of the counting numbers from 1 to 500, the sum will be 125520. If you don't believe me then just check my answer by doing it the long way.

Have fun!

Answer 7

1. group the numbers in pairs. e.g. (1+500), (2+499), (3+498)...each pair would equal to 501.

2. then take 500/2 since there are 2 numbers in a pair...

3. 250*501= 125250. tada! :)

Answer 8

the formula for the sum of conceutive intergers 1 through n is n(n+1)/2.......so.....500(500+1)/2= (25,0000+500)/2=125,250

btw the formula for consecutive integers to the 2nd power is n(n+1)(2n+1)/6

Answer 9

you mean... 1 + +2 +3 + --- + 500?

(500*501)/2 = 250*501
= 125250

Answer 10

n
Σ k = n(n+1) / 2 = 1/2 (n+1)P2
k =0
500 * 501 / 2 = 125 25