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$]$
$=\frac{7}{9} [(10 - 1 ) + ( 10^2 -1) + ( 10^3 - 1) + ( 10^4 - 1)$
$\qquad \qquad + ... n$ terms ]
$ = \frac{7}{9} [(10 + 10^2 + 10^3 + ... n$ terms)
$\qquad \qquad - ( 1 + 1 + 1 + ... n$ 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]$.