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Prove that the $n^{th}$ term of an A.P. can't be $n^2+1$.
Given: The $n^{th}$ term of a sequence is $n^2+1$.
To do: To prove that the $n^{th}$ term of an A.P. can't be $n^2+1$.
Solution:
Taking $n=1$ in $n^2+1$, we get
$a_1=1^2+1=1+1=2$
Taking $n=2$, we get
$a_2=2^2+1=4+1=5$
Taking $n=3$, we get
$a_3=3^2+1=9+1=10$
$a_2-a_1=5-2=3$
And $a_3-a_2=10-5=5$
Thus, we observe that the common difference is not the same.
Hence, $n^{th}$ term of an AP cannot be $n^2+1$.
Updated on: 10-Oct-2022
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