The sum of first n terms of A.P. is $3 n + n^{2}$ then Find second, third and $15^{t h}$ term.

6

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8

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32

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45

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Solution

The correct options are
B $6$
C $8$
D $32$
Here $S_{n} = 3 n + n^{2}$
Now, first term $a = S_{1} \Rightarrow a = 3 \times 1 + 1^{2} = 4$ and
Sum of first two terms= $a_{1} + a_{2} = a + (a + d) = S_{2} = 3 \times 2 + 2^{2} = 10$
$\Rightarrow 2 a + d = 10 \Rightarrow 8 + d = 10 \Rightarrow d = 2$
Now, $a_{2} = a + d \Rightarrow a_{2} = 4 + 2 = 6$
$a_{3} = a + 2 d \Rightarrow a_{3} = 4 + 2 \times 2 = 8$
$a_{15} = a + 14 d \Rightarrow a_{15} = 4 + 14 \times 2 = 32$

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