The sum of 1 + 2 5 + 3 5^2 + 4 5^3 + ........... upto n terms is - Vidyadip

MCQ

The sum of $1 + \frac{2}{5} + \frac{3}{{{5^2}}} + \frac{4}{{{5^3}}} + ...........$ upto $n$ terms is

✓
$\frac{{25}}{{16}} - \frac{{4n + 5}}{{16 \times {5^{n - 1}}}}$
B
$\frac{3}{4} - \frac{{2n + 5}}{{16 \times {5^{n + 1}}}}$
C
$\frac{3}{7} - \frac{{3n + 5}}{{16 \times {5^{n - 1}}}}$
D
$\frac{1}{2} - \frac{{5n + 1}}{{3 \times {5^{n + 2}}}}$

✓

Answer

Correct option: A.

$\frac{{25}}{{16}} - \frac{{4n + 5}}{{16 \times {5^{n - 1}}}}$

a
(a) Given series, let ${S_n} = 1 + \frac{2}{5} + \frac{3}{{{5^2}}} + \frac{4}{{{5^3}}} + ......... + \frac{n}{{{5^{n - 1}}}}$

$\frac{1}{5}{S_n} = {\rm{ }}\frac{1}{5} + \frac{2}{{{5^2}}} + \frac{3}{{{5^3}}} + ....... + \frac{n}{{{5^n}}}$

Subtracting,

$\left( {1 - \frac{1}{5}} \right){S_n} = 1 + \frac{1}{5} + \frac{1}{{{5^2}}} + \frac{1}{{{5^3}}} + ...... + {\rm{upto}}\;n\;{\rm{terms}}\; - \frac{n}{{{5^n}}}$

$ \Rightarrow $$\frac{4}{5}{S_n} = \frac{{1 - \frac{1}{{{5^n}}}}}{{\frac{4}{5}}} - \frac{n}{{{5^n}}}$

$ \Rightarrow $${S_n} = \frac{{25}}{{16}} - \frac{{4n + 5}}{{16 \times {5^{n - 1}}}}$.

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