Question

Find the sum of the sequence 7, 77,777, 7777.......... to n terms.
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Answer 1

Answer:

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Answer 2

EXPLANATION.

$\sf : \implies \: 7 + 77 + 777 + ...... \: n \: terms \: \\ \\ \sf : \implies \: 7(1 + 11 + 111 + ..... \: n \: terms) \\ \\ \sf : \implies \: \frac{7}{9} (9 + 99 + 999 + ....... \: n \: terms) \\ \\ \sf : \implies \: \frac{7}{9}[(10 - 1) + (100 - 1)(1000 - 1) + ..... \: n \: terms] \\ \\ \sf : \implies \: \frac{7}{9}[10 + 100 + 1000 + .... \: n \: terms \: - (1 + 1 + 1 + .... \: n \: terms)]$

$\sf : \implies \: \dfrac{7}{9}[10 + 100 + 1000 + .... \: n \: terms \: - \: n] \\ \\ \sf : \implies \: first \: term \: = 10 \\ \\ \sf : \implies \: common \: ratio \: = r \: = 10 \\ \\ \sf : \implies \: sum \: of \: n \: terms \: of \: gp \implies \: s_{n} \: = \frac{a( {r}^{n} - 1)}{r - 1}$

$\sf : \implies \: s_{n} \: = \frac{7}{9} [ \dfrac{10(10 {}^{n} - 1)}{9} - n ] \\ \\ \sf : \implies \: s_{n} \: = \frac{70}{81} [10 {}^{n} - 1] - n \:$