Question

What Is the sum of all the odd numbers from 1 to 1000

Answer 1

How many numbers would that be?

Answer 2

\[1 + 3 + 5+7+9+ ... + 999\]Add every number by one.\[(1 + 3 + 5+7+9+ ... + 999) + 500 = (1+1) + (3+1) + (5+1) + (7+1) + ... + (999 + 1)\]\[= 2 + 4 + 6 + 8 + ... + 1000\]
Now divide all of this by two:\[\frac{2 + 4 + 6 + 8 + ... + 1000}{2}= 1 + 2 + 3 + 4 + 5 + ... + 500 = \sum_{n = 1}^{500}n = \frac{n(n+1)}{2}\]\[= \frac{500(501)}{2} = 250(501) = \boxed{125250}\]

Since we changed the sequence by adding 500 and divide by two, we have to add all of this by multiply this by two then subtract 500.

2(125250) - 500 = ?

Does this make sense?

Answer 3

There is a much easier way to do that.

Answer 4

***Since we changed the sequence by adding 500 and divide by two, we have to multiply this by two then subtract 500 to get what we want.

Sorry about confusion. My brain is dead, I guess I need to take some break times, lol.

Answer 5

@CliffSedge Can you show it to us?

Answer 6

There are 500 odd numbers from 1 to 1000: 1, 3, 5, . . . 997, 999.

Answer 7

The first and last terms 1+999 = 1000, so do the second and second-to-last, 3+997=1000.

Answer 8

There are 250 pairs of numbers that sum to 1000.
250×1000 = 250,000.

Answer 9

@CliffSedge

Answer 10

:-D