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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.
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