Question

Use​ Gauss's approach to find the following sums​ (do not use​ formulas). 1+3+5+7+...999

Answer 1

`S=1+3+5+\cdots+995+997+999`

`S=999+997+995+\cdots+5+3+1`

`\implies 2S=(1+999)+(3+997)+(5+995)+\cdots+(995+5)+(997+3)+(999+1)`


`2S=\underbrace{1000+1000+\cdots+1000+1000}_{500\text{ times}}`

(We are adding 500 instances of 1000 because that's how many odd numbers there are between 1 and 1000, inclusive.)


`2S=500*1000=500000`

`\implies S=250000`