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 terms as

$s_n$ = 8 + 88 + 888 + 8888 + …………….. to n terms

=$\frac{8}{9}$ [9 + 99 + 999 + 9999 + ........ to n terms]

=$\frac{8}{9}$ [ (10 - 1) + ($10^2$ - 1) + ($10^3$ - 1) + ($10^4$ - 1) + ........ to n terms]

= $\frac{8}{9}$ [(10 + $10^2$ + ....... n terms) - (1 + 1 + 1 + ..... n terms)]

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

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

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