Question

find the sum to n terms of the sequence 8,88,888,8888...........​

Answer 1

$\small \star \underline \bold \red{given}$

$\huge \underline \bold \green{the \: given \: sequence \: is \: 8 \: 88 \: 888 \: 8888}$

$\huge \underline \bold \green{this \: sequence \: is \: not \: a \: g.p. \: however \: i \: can \: be \: changed \: to \: gp \: by \: writing \: the \: terms \: as}$

$s \: 8 + 88 + 888 + 8888$

$\frac{8}{9} |9 + 99 + 999 + 9999 \: + ............to \: n \: terms|$

$\frac{8}{9 |(10 - 1)( {10}^{2} - 1( {10}^{3} - 1) ( {10}^{4} - 1) + to \: n \: terms | }$

$\frac{8}{9} |(10 + {10}^{2} + ......n \: terms| - (1 + 1 + 1 + n \: terms)$

$\frac{8}{9} | \frac{10( {10}^{n}) }{10 - 1} - n |$

$\frac{8}{9} | \frac{10( {10}^{n } - 1)}{9} - n |$

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