A) \[\frac{16}{35}\]
B) \[\frac{11}{8}\]
C) \[\frac{35}{16}\]
D) \[\frac{7}{16}\]
Correct Answer: C
Solution :
Given, \[1+\frac{4}{5}+\frac{7}{{{5}^{2}}}+\frac{10}{{{5}^{3}}}+...\,\text{to}\,\infty \] This sequence is arithmetico-geometric sequence. Then, \[{{S}_{\infty }}=\frac{a}{1-r}+\frac{dr}{{{(1-r)}^{2}}}\] Here, \[a=1,r=\frac{1}{5},d=3\] \[\therefore \] \[{{S}_{\infty }}=\frac{1}{1-\frac{1}{5}}+\frac{3\times \frac{1}{5}}{{{\left( 1-\frac{1}{5} \right)}^{2}}}\] \[=\frac{5}{4}+\frac{3}{5\times \frac{16}{25}}=\frac{5}{4}+\frac{15}{16}=\frac{35}{16}\] Note: The sum of infinite arithmetico-geometric sequence is \[{{S}_{\infty }}=\frac{a}{1-r}+\frac{dr}{{{(1-r)}^{2}}},\] where d is common difference, r is common ratio and a is first term.
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