Evaluate $7 + 77 + 777 + . . . . . . . . . . . . .$ upto $n$ terms.

\frac{7}{81} [10^{n + 1} - 9 n]

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\frac{7}{81} [10^{n + 1} - 9 n - 10]

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\frac{7}{81} [10^{n} - 9 n - 10]

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\frac{7}{81} [10^{n + 1} - n - 10]

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Solution

The correct option is D $\frac{7}{81} [10^{n + 1} - 9 n - 10]$
Let $S = 7 + 77 + 777 + . . . . . . . . +$ up to $n$ terms
$\Rightarrow \frac{7}{9} [9 + 99 + 999 + . . . . .]$

$\Rightarrow \frac{7}{9} [(10 - 1) + (10^{2} - 1) + (10^{3} + 1) + . . . . . . . . u p t o n t e r m s]$

$\Rightarrow \frac{7}{9} [10 + 10^{2} + 10^{3} + . . . . . . + 10^{n} - n]$

$\Rightarrow \frac{7}{9} [10 \frac{10^{n} - 1}{9} - n]$

$\Rightarrow \frac{7}{81} [10^{n + 1} - 9 n - 10]$

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Similar questions

\underset{n - digits}{. . . . . . . . . . .} 7 = \frac{7}{81} (10^{n + 1} - 9 n - 10)

Prove that $7 + 77 + 777 + . . . . . . + 777 . . . . . . . . . . . . . n - d i g i t s 7 = \frac{7}{81} (10^{n + 1} - 9 n - 10)$

\frac{9^{n}}{10^{n}} < \frac{1}{2}

n

8 + 88 + 888 + . . . . . . . . .

\frac{m}{81} [k^{n + 1} - 9 n - k]

k - m

lim n \to \infty [\frac{7}{10} + \frac{29}{10^{2}} + \frac{133}{10^{3}} + \dots \dots + \frac{5^{n} + 2^{n}}{10^{n}}]

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