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The sum of the first n terms of an A.P. is given by $S n = 2 n 2 + 5 n$ , then the $n^{th}$ term of the A.P is ____.

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Detailed Solution

The sum of the first n terms of an A.P. is given by

S n = 2 n 2 + 5 n

, then the

n^{th}

term of the A.P is

3 + 4 n

.
Given that,
The sum of the first n terms of an A.P. is given by

S n = 2 n 2 + 5 n

We know,
An arithmetic progression is given by

a n = a + n − 1 d

.
The first term is a.
The common difference is d.
The number of terms are n.
For

n = 1

S 1 = 212 + 51 ⇒ S 1 = 2 + 5 ⇒ S 1 = 7

Therefore the first term is,

a = 7

.
For

n = 2

S 2 = 222 + 52 ⇒ S 2 = 8 + 10 ⇒ S 2 = 18

Therefore the second term is,

a 2 = S 2 − S 1 ⇒ a 2 = 18 − 7 ⇒ a 2 = 11

For

n = 3

S 3 = 232 + 53 ⇒ S 3 = 18 + 15 ⇒ S 3 = 33

Therefore the third term is,

a 3 = S 3 − S 2 ⇒ a 3 = 33 − 18 ⇒ a 3 = 15

Thus the arithmetic progression is, 7,11,15,... .
Here,

a = 7 d = 4

Therefore,

a n = 7 + n − 1 4 ⇒ a n = 7 + 4 n − 4 ⇒ a n = 3 + 4 n

Therefore, the

n t h

term is

3 + 4 n

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