Question

$\huge \mathfrak \color{green}{question} :$

Find the sum of the sequence 6, 66, 666, 6666, ... to n terms.​

Answer 1

$\huge \color{green}\underbrace{\color{red}Answer}:$

$\\ \\ \large \sf \color{brown} \implies \frac{2}{3}[ \frac{10(10^{n}-1}{9}-n]$

$\huge \color{green} \underbrace{\color{red}{Solution}}:$

Sn = 6+66+666+6666+... to n terms

$\ \ \ \sf Sn = \frac{6}{9}[9+99+999+9999+...to \: n \: terms ] \\ \\ \sf \ \ \ \ = \frac{2}{3}[(10-1)+(100-1)+(1000-1)+(10000-1) +... to \ n \ terms ] \\ \\ \sf \ \ \ = \frac{2}{3}[(10+10²+10³+10⁴+...n \: terms) -(1+1+1+... n \: terms )] \\ \\ \sf \ \ \ \ = \frac{2}{3}[\frac{10(10^{n}-1}{10-1}-n] \\ \\ \sf \ \ \ \ \ \ = \boxed{ \sf \frac{2}{3}[ \frac{10(10^{n}-1}{9}-n] }$

$\color{red}━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━$