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The sum of the first terms of an A.P. is $S n = 3 n 2 + 2 n$ . What will be its 15th term?

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answer is B.

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Detailed Solution

Given that the sum of the first n terms of an A.P is

S n = 3 n 2 + 2 n

.
The nth term of an arithmetic progression with first term ‘a’ and common difference ‘d’ is

a n = a + n − 1 d

.
Taking the values for

n

from 1,2,3... and substituting them in the given sum,
When

n = 1

⇒ S 1 = 312 + 21 ⇒ S 1 = 3 + 2 ⇒ S 1 = 5

Then,

a 1 = S 1 ⇒ a 1 = 5

The first term is

a = 5

.
When

n = 2

⇒ S 2 = 322 + 22 ⇒ S 2 = 12 + 4 ⇒ S 2 = 16

Then,

a 2 = S 2 − S 1 ⇒ a 2 = 16 − 5 ⇒ a 2 = 11

The second term is

a 2

= 16.
When

n = 3

⇒ S 3 = 332 + 23 ⇒ S 3 = 27 + 6 ⇒ S 3 = 33

Then,

a 3 = S 3 − S 2 ⇒ a 3 = 33 − 16 ⇒ a 3 = 17

The third term is

a 3

= 17.
Therefore, the arithmetic progression will be 5,11,17,... .
Here,

a = 5 d = a 2 − a 1 ⇒ d = 11 − 5 ⇒ d = 6

Substituting a=5, n=15 and d=

6

in the equation

a n = a + n − 1 d

⇒ a 15 = 5 + 15 − 1 6 ⇒ a 15 = 5 + (14 × 6) ⇒ a 15 = 5 + 84 ⇒ a 15 = 89

Thus the 15th term will be 89.
Therefore the correct option is 2.

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The sum of the first terms of an A.P. is S n =3 n 2 +2n . What will be its 15th term?

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The sum of the first terms of an A.P. is $S n = 3 n 2 + 2 n$ . What will be its 15th term?