The $n^{t h}$ term of a sequence is given by $a_{n} = 2 n^{2} + n + 1$ . Show that it is not an A.P.

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Solution

$a_{n} = 2 n^{2} + n + 1$

$a_{1} = 2 (1)^{2} + (1) + 1 = 4$

$a_{2} = 2 (2)^{2} + (2) + 1 = 11$

$a_{3} = 2 (3)^{3} + (3) + 1 = 21$

Since, $a_{3} - a_{2} \neq a_{2} - a_{1}$

$∴$ The given sequence is not as A.P. as consecutive terms do not have a common difference.

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