First slide

Geometric progression

Question

The sum of the series 5 +55 +555 +...to n terms is

Easy

A

$\frac{1}{80} (10^{n + 1})$
B

$\frac{5}{81} (10^{n + 1} - 9 n - 10)$
C

$\frac{5}{81} (10^{n - 1} - 8 n - 1)$
D

$\frac{5}{81} (10^{n + 1} + 9 n + 10)$

Solution

5 + 55 + 555 +...to n terms
=5 {1+ 1 1 + 1 1 1 +...to n terms}
$= \frac{5}{9} {9 + 99 + 999 + \dots to n terms}$
[multiply numerator and denominator by 9]
$\begin{array}{l} = \frac{5}{9} {(10 - 1) + (100 - 1) + (1000 - 1) + \dots to n terms} \\ = \frac{5}{9} {10 + 100 + 1000 + \dots + n terms - (1 + 1 + \dots + n terms)} \\ = \frac{5}{9} \{\frac{10 (10^{n} - 1)}{10 - 1} - n\} \\ = \frac{5}{81} \{10^{n + 1} - 10 - 9 n\} \end{array}$

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