Question

Find the sum of the following numbers without actually adding the numbers.
(i) $1+3+5+7+9+11+13+15$
(ii) $1+3+5+7$
(iii) $1+3+5+7+9+11+13+15+17$

Answer 1

Given $S=1+3+5+7+....+(2n-1)$

This is an arithmetic progression

Sum of AP: $S_n=\frac{n}{2}[2a+(n-1\left)d\right]=\frac{n}{2}[2+(n-1\left)2\right]=n^2$

$\left(i\right)$ $1+3+5+7+9+11+13+15$

$n=8S_n=n^2=8^2=64$

$\left(ii\right)$ $1+3+5+7$

$n=4S_n=n^2=4^2=16$

$\left(iii\right)$ $1+3+5+7+9+11+13+15+17$

$n=9S_n=n^2=9^2=81$