Question

Find the sum of the sequence $7,77,777,7777,...$ to $n$ terms.

• A. $\frac{7}{9}\left[\frac{10\left(10^n-1\right)}{9}-n\right]$
• B. $\frac{7}{9}\left[\frac{10\left(10^n+1\right)}{9}-n\right]$
• C. $\frac{10^n-1}{9}$
• D. None of these

Answer 1

Answer: (A)

This is not a $G.P$., however, we can relate it to a $G.P$. by writing the terms as
$S_n=7+77+777+7777+...$ to $n$ terms
$=\frac{7}{9}[9+99+999+9999+...$ to $n$ terms$]$
$=\frac{7}{9}[(10-1)+(10^2-1)+(10^3-1)+(10^4-1)$
$+...n$ terms ]
$=\frac{7}{9}[(10+10^2+10^3+...n$ terms)
$-(1+1+1+...n$ terms)]

$=\frac{7}{9}\left[\frac{10\left(10^n-1\right)}{10-1}-n\right]$

$=\frac{7}{9}\left[\frac{10\left(10^n-1\right)}{9}-n\right]$.