The sum of 1 + 25 + 352 + 453 + .............upto n terms is

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Question

The sum of 1 +

\frac{2}{5}

+

\frac{3}{5^{2}}

+

\frac{4}{5^{3}}

+ .............upto n terms is

Options:

A . 2516 - 4n+516×5n−1

B . 34 - 2n+516×5n+1

C . 37 - 3n+516×5n−1

D . 12 - 5n+13×5n+2

Answer: Option A
:
A
Given series, let

S_{n}

= 1 +

\frac{2}{5}

+

\frac{3}{5^{2}}

+

\frac{4}{5^{3}}

+ .............+

\frac{n}{5^{n - 1}}

\frac{1}{5}

S_{n}

=

\frac{1}{5}

+

\frac{2}{5^{2}}

+

\frac{3}{5^{3}}

+ ..............+

\frac{n}{5^{n}}

Subtracting,
(1 -

\frac{1}{5}

)

S_{n}

= 1 +

\frac{1}{5}

+

\frac{1}{5^{2}}

+

\frac{1}{5^{3}}

+ ............+ upto n terms -

\frac{n}{5^{n}}

⇒

\frac{4}{5}

S_{n}

=

\frac{1 - \frac{1}{5^{n}}}{\frac{4}{5}}

-

\frac{n}{5^{n}}

⇒

S_{n}

=

\frac{25}{16}

-

\frac{4 n + 5}{16 \times 5^{n - 1}}

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