A) \[\frac{3n(n+1)}{2}\]
B) \[\frac{{{n}^{2}}(n+1)}{2}\]
C) \[\frac{n{{(n+1)}^{2}}}{4}\]
D) \[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}\]
Correct Answer: B
Solution :
The sum of n terms of given series\[=\frac{n{{(n+1)}^{2}}}{2},\]if n is even. Let n is odd ie., \[n=2m+1\] Then,\[{{S}_{2m+1}}={{S}_{2m}}+(2m+1)\]th term \[=\frac{(n-1){{n}^{2}}}{2}+nth\text{ }term\] \[=\frac{(n-1){{n}^{2}}}{2}+{{n}^{2}}\] \[(\because n\,is\,odd=2m+1)\] \[={{n}^{2}}\left[ \frac{n-1+2}{2} \right]=\frac{(n+1){{n}^{2}}}{2}\]
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