What is the sum of the first 10 terms of the | vedclass

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(B) Let the sum be $S = 0.7 + 0.77 + 0.777 + \dots$ up to $10$ terms.We can write this as $S = 7(0.1 + 0.11 + 0.111 + \dots)$.Multiply and divide by $

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$\frac{7}{81} \left( 89 + \frac{1}{10^{10}} \right)$

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(B) Let the sum be $S = 0.7 + 0.77 + 0.777 + \dots$ up to $10$ terms.We can write this as $S = 7(0.1 + 0.11 + 0.111 + \dots)$.Multiply and divide by $9$: $S = \frac{7}{9} (0.9 + 0.99 + 0.999 + \dots)$.This can be rewritten as $S = \frac{7}{9} [(1 - 0.1) + (1 - 0.01) + (1 - 0.001) + \dots]$.For $10$ terms,$S = \frac{7}{9} [10 - (0.1 + 0.01 + \dots + 0.1^{10})]$.The sum inside the bracket is a geometric progression with $a = 0.1$,$r = 0.1$,and $n = 10$.Sum $= \frac{a(1 - r^n)}{1 - r} = \frac{0.1(1 - 0.1^{10})}{1 - 0.1} = \frac{0.1(1 - 10^{-10})}{0.9} = \frac{1}{9} (1 - 10^{-10})$.Substituting this back: $S = \frac{7}{9} [10 - \frac{1}{9} (1 - 10^{-10})] = \frac{7}{9} [\frac{90 - 1 + 10^{-10}}{9}] = \frac{7}{81} (89 + 10^{-10})$.Thus,the sum is $\frac{7}{81} (89 + \frac{1}{10^{10}})$.

What is the sum of the first $10$ terms of the series $0.7 + 0.77 + 0.777 + \dots$?

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