Find the sum of the following to n terms: 7 + 77 + 777 + 7777 + ………

$S_n = 7 + 77 + 777 + 7777 + ……… \text{ upto n terms} \\[1em] = 7(1 + 11 + 111 +…..) \text{ upto n terms} \\[1em] = \dfrac{7}{9} (9 + 99 + 999 +…..) \text{ upto n terms} \\[1em] = \dfrac{7}{9}[(10 - 1) + (10^2 - 1) + (10^3 - 1)+…….+(10^n - 1)] \\[1em] = \dfrac{7}{9}[(10 + 10^2 + 10^3+…..+10^n) - (1 + 1 + 1+…..\text{n times})] \\[1em] = \dfrac{7}{9}[(10 + 10^2 + 10^3+…..+10^n) - n] \text{ …….(1)}$

Now,

Calculating the sum of 10 + 10² + 10³ + ……… + 10ⁿ.

a = 10

r = $\dfrac{10^2}{10}$ = 10

We know that,

The sum of the first n terms of a G.P. is given by:

$S_n = \dfrac{a(r^n - 1)}{r - 1}$ [For r > 1]

$\Rightarrow S_n = \dfrac{10(10^n - 1)}{10 - 1} \\[1em] = \dfrac{10(10^n - 1)}{9}.$

Substitute and Simplify, the above value in equation (1), we get :

$S_n = \dfrac{7}{9}\Big[\dfrac{10(10^n - 1)}{9} - n\Big] \\[1em] = \dfrac{7}{81}\Big[10 \times 10^n - 10 - 9n\Big] \\[1em] = \dfrac{7}{81}\Big[10^{n + 1} - 9n - 10\Big]$

Hence, S_n = $\dfrac{7}{81}\Big[10^{n + 1} - 9n - 10\Big]$.