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 3 + 33 + 333 +?......+ n terms Given series can be written as, \[=\frac{1}{3}[9+99+999+........+n\,\,\text{terms }\!\!]\!\!\text{ }\] \[=\frac{1}{3}\left[ (10-1)+({{10}^{2}}-1)+({{10}^{3}}-1)+....+n\,\text{terms} \right]\] \[=\frac{1}{3}\left[ 10+{{10}^{2}}+....+{{10}^{n}} \right]\]\[-\frac{1}{3}\left[ 1+1+1+....+n\,\text{terms} \right]\] \[=\frac{1}{3}\,.\,\frac{10\,({{10}^{n}}-1)}{10-1}-\frac{1}{3}.n\,\]\[=\frac{1}{3}\left[ \frac{{{10}^{n+1}}-10}{9}-n \right]\] \[=\frac{1}{3}\,\left[ \frac{{{10}^{n\,+\,1}}-9n-10}{9} \right]\]\[=\frac{1}{27}[{{10}^{n\,+\,1}}-9n-10]\].
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