Show that sequence of following $n^{t h}$ terms ,are not in A.P.
$n^{2} + 1$

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Solution

The first three terms can be calculated by substituting $n = 1, 2, 3$
Let them be denoted by $a, b, c$ respectively.
$a = 2$
$b = 5$
$c = 10$
$a + c = 12$
$2 b = 10$
If $a, b, c$ actually were in an A.P., then $a + c$ would be equal to $2 b$ which is not the case here. Hence, the given sequence in not an A.P.

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