Question

Find the sum to $n$ terms of the sequence $8,88,888,8888,....$ .

Answer 1

The given sequence is $8,88,888,8888....$

This sequence is not a G.P. However,it can be changed to G.P. by writing the term as

$S_n=8+88+888+8888+......+nterms$

$=\frac{8}{9}[9+99+999+9999+....tonterms]$

$=\frac{8}{9}\left[\right(10-1)+({10}^2-1+({10}^3-1)+({10}^4-1+....tonterms]$

$=\frac{8}{9}\left[\right(10+{10}^2+......nterms)-(1+1+1+...nterms\left)\right]$

$=\frac{8}{9}[\frac{10({10}^n-1)}{10-1}-n]$ [Sum of GP$=\frac{a(r^n-1)}{r-1}$ when $r>1$]

$=\frac{8}{9}[\frac{10({10}^n-1)}{9}-n]$

$=\frac{80}{81}({10}^n-1)-\frac{8}{9}n$