Question

0=1+3+5+7+...+101 find the sum
Without computing

Answer 1

Let

S = 1 + 3 + 5 + ... + 97 + 99 + 101

We can reverse the order of terms to write this as

S = 101 + 99 + 97 + ... + 5 + 3 + 1

Then adding these together gives

2S = (1 + 101) + (3 + 99) + (5 + 97) + ...

2S = 102 + 102 + 102 + ...

where there are 51 copies of 102 on the right side, so

2S = 51 * 102 ==> S = 5202