Question

Find the sum to $(2n+1)$ terms of the sequence : $1,-1,1,-1,1,-1,1,-\mathrm{1......}$

Answer 1

$1,-1,+1,-1,1,-1--------(2n+1)$ term

$S_n=1+(-1)+1+(-1)+----+(2n+1)$ term

above given expression is Geometric progression

where

first term = $1\leftarrow$ common ratio = -1.

now, total term = 2n+1.

sum of n term of GP = $\frac{a(r^n-1)}{r-1}$ or $\frac{a(1-r^n)}{1-r}$

$S_{2n+1}=\frac{1[1-(-1{)}^{2n+1}]}{1-(-1)}=1\left[\frac{1-(-1)}{1+1}\right]=1$

$\therefore$ sum of $2n+1$ terms of given sequence is 1.