A) \[{{n}^{3}}+{{n}^{2}}+1\]
B) \[{{n}^{3}}-{{n}^{2}}+1\]
C) \[{{n}^{3}}-{{n}^{2}}\]
D) \[{{n}^{3}}+{{n}^{2}}\]
Correct Answer: B
Solution :
[b] Given, \[{{a}_{0}}=1,{{a}_{n+1}}=3{{n}^{2}}+n+{{a}_{n}}\] |
\[\Rightarrow {{a}_{1}}=3(0)+0+{{a}_{0}}=1\] |
\[\Rightarrow {{a}_{2}}=3{{(1)}^{2}}+1+{{a}_{1}}=3+1+1=5\] |
For option (b), |
Let \[P(n)={{n}^{3}}-{{n}^{2}}+1\] |
\[\therefore \,\,\,\,\,P(0)=0-0+1=1={{a}_{0}}\] |
\[P(1)={{1}^{3}}-{{1}^{2}}+1=1={{a}_{1}}\] |
and \[P(2)={{(2)}^{3}}-{{(2)}^{2}}+1=5={{a}_{2}}\] |
\[\therefore \,\,\,\,\,{{a}_{n}}={{n}^{3}}-{{n}^{2}}+1\] |
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