A) \[{{n}^{2}}(n+1)\]
B) \[\frac{{{n}^{2}}(n-1)}{2}\]
C) \[\frac{{{n}^{2}}(n+1)}{2}\]
D) \[{{n}^{2}}(n-1)\]
Correct Answer: C
Solution :
If n is odd, the required sum is \[{{1}^{2}}+{{2.2}^{2}}+{{3}^{2}}+{{2.4}^{2}}+....+2{{(n-1)}^{2}}+{{n}^{2}}\] \[=\frac{\left( n-1 \right){{\left( n-1+1 \right)}^{2}}}{2}+{{n}^{2}}\left( \because n-1\,\text{is}\,\text{even} \right)\] \[=\left( \frac{n-1}{2}+1 \right){{n}^{2}}=\frac{{{n}^{2}}(n+1)}{2}\]
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