A) \[\frac{1}{27}({{10}^{n+1}}+9n-28)\]
B) \[\frac{1}{27}({{10}^{n+1}}-9n-10)\]
C) \[\frac{1}{27}({{10}^{n+1}}+10n-9)\]
D) None of these
Correct Answer: B
Solution :
\[Series\text{ }3\text{ }+\text{ }33\text{ }+\text{ }333+.....+\,\,n\,\,terms\] Given series can be written as, \[=\,\,\,\frac{1}{3}\left[ 9+99+999+....+n terms \right]\] \[= \frac{1}{3}[\left( 10 -1 \right) + \left( 100-1 \right)+\left( 1000-1)+....+m\,\,terms \right]\] \[=\,\,\frac{1}{3}\left[ 10+{{10}^{2}}+....+{{10}^{n}} \right]-\frac{1}{3}\left[ 1+1+1+....+n\,\,terms \right]\]\[=\,\,\frac{1}{3}.\frac{10({{10}^{n}}-1)}{10-1}-\frac{1}{3}n=\frac{1}{3}\left[ \frac{{{10}^{n+1}}-10}{9}-n \right]\] \[=\,\,\frac{1}{3}.\left[ \frac{{{10}^{n+1}}-9n-10}{9} \right]=\,\,\frac{1}{27}\,\left[ {{10}^{n+1}}-9n-10 \right]\]
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