Question

Use​ Gauss's approach to find the following sums​ a. 1+2+3+4..+999 b. 1+3+5..+997

Answer 1

$S=1+2+3+4+\cdots+996+997+998+999$
$S=999+998+997+996+\cdots+4+3+2+1$
$\implies 2S=(1+999)+(2+998)+\cdots+(998+2)+(999+1)$

Note that we're adding 999 terms in $S$, so $2S$ is summing together 999 pairs of numbers that add to 1000:

$2S=1000+1000+\cdots+1000+1000$
$2S=999(1000)$
$S=\dfrac{999(1000)}2=999(500)=499500$

$T=1+3+5+\cdots+993+995+997$
$T=997+995+993+\cdots+5+3+1$
$\implies 2T=(1+997)+(3+995)+\cdots+(995+3)+(997+1)$

This time, $T$ adds up 499 numbers - we can determine this by finding the value of $n$ such that $2n-1=997$ - that each add up to 998, so

$2T=998+998+\cdots+998+998$
$2T=499(998)$
$T=\dfrac{499(998)}2=499(499)=249001$