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Question
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
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Solution
We have,
7 + 77 + 777 + ... n terms
\[S_n\] = 7 [1 + 11 + 111 + ... n terms]
\[= \frac{7}{9}\left( 9 + 99 + 999 + . . . \text { n terms } \right)\]
\[ = \frac{7}{9}\left\{ \left( 10 - 1 \right) + \left( {10}^2 - 1 \right) + \left( {10}^3 - 1 \right) + . . . + \left( {10}^n - 1 \right) \right\}\]
\[ = \frac{7}{9}\left\{ \left( 10 + {10}^2 + {10}^3 + . . . + {10}^n \right) \right\} - \left( 1 + 1 + 1 + 1 . . . \text {n times }\right)\]
\[ = \frac{7}{9}\left\{ 10 \times \frac{\left( {10}^n - 1 \right)}{10 - 1} - n \right\} = \frac{7}{9} \left\{ \frac{10}{9}\left( {10}^n - 1 \right) - n \right\}\]
\[ = \frac{7}{81}\left\{ {10}^{n + 1} - 9n - 10 \right\}\]
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