Justify whether it is true to say that the following are the $ n^{\text {th }} $ terms of an AP.
$ 3 n^{2}+5 $

Given:

$a_n = 3n^2 + 5$

To do:

We have to justify whether it is true to say that \( 3 n^{2}+5 \) is the \( n^{\text {th }} \) term of an AP.

Solution:

To  check whether the sequence defined by $a_n = 3n^2 + 5$ is an A.P., we have to check whether the difference between any two consecutive terms is equal.

Let us find the first few terms of the sequence by substituting $n=1, 2, 3....$

When $n=1$,

$a_1=3(1)^2+5$

$=3+5$

$=8$

$a_2=3(2)^2+5$

$=3(4)+5$

$=17$

$a_3=3(3)^2+5$

$=3(9)+5$

$=32$

$a_4=3(4)^2+5$

$=3(16)+5$

$=53$

Here,

$a_2-a_1=17-8=9$

$a_3-a_2=32-17=15$

$d=a_4-a_3=53-32=21$

$a_2-a_1≠a_3-a_2≠a_4-a_3$

Hence, the given sequence is not an A.P.