A) \[\frac{25}{16}-\frac{4n+5}{16\times {{5}^{n-1}}}\]
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) \[\frac{1}{2}-\frac{5n+1}{3\times {{5}^{n+2}}}\]
Correct Answer: A
Solution :
Given series, let \[{{S}_{n}}=1+\frac{2}{5}+\frac{3}{{{5}^{2}}}+\frac{4}{{{5}^{3}}}+.........+\frac{n}{{{5}^{n-1}}}\] \[\frac{1}{5}{{S}_{n}}=\text{ }\frac{1}{5}+\frac{2}{{{5}^{2}}}+\frac{3}{{{5}^{3}}}+.......+\frac{n}{{{5}^{n}}}\] Subtracting, \[\left( 1-\frac{1}{5} \right){{S}_{n}}=1+\frac{1}{5}+\frac{1}{{{5}^{2}}}+\frac{1}{{{5}^{3}}}+......+\text{upto}\ n\ \text{terms}\ -\frac{n}{{{5}^{n}}}\] \[\Rightarrow \]\[\frac{4}{5}{{S}_{n}}=\frac{1-\frac{1}{{{5}^{n}}}}{\frac{4}{5}}-\frac{n}{{{5}^{n}}}\]\[\Rightarrow \]\[{{S}_{n}}=\frac{25}{16}-\frac{4n+5}{16\times {{5}^{n-1}}}\].
You need to login to perform this action.
You will be redirected in
3 sec
