A) \[{{n}^{2}}+\frac{4}{9}\left( 1+\frac{1}{{{10}^{n}}} \right)\]
B) \[{{n}^{2}}+\frac{4}{9}\left( 1-\frac{1}{{{10}^{n}}} \right)\]
C) \[n+\frac{4}{9}\left( 1-\frac{1}{{{10}^{n}}} \right)\]
D) None of these
Correct Answer: B
Solution :
\[1.4+3.04+5.004+...\] \[=(1+3+5+...)+(0.4+0.04+0.004+...)\] \[=\frac{n}{2}[2+(n-1)\cdot 2]+\left[ \frac{4}{10}+\frac{4}{{{10}^{2}}}+...\,\,upto\,\,n\,\,terms \right]\]\[={{n}^{2}}+\frac{4}{10}\left[ \frac{1-{{\left( \frac{1}{10} \right)}^{n}}}{1-\frac{1}{10}} \right]\] \[={{n}^{2}}+\frac{4}{9}\left( 1-\frac{1}{{{10}^{n}}} \right)\]
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