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In an AP, if $S n = n (4 n + 1),$ then the AP is

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2,4,6,8

5,13,21,29

24,28,32,36

29,21,13,5

answer is B.

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Detailed Solution

The given sum equation of an A.P series is

S n = n 4 n + 1

.
Using the above equation we will find the terms of the series.

a 1 = S 1, a 2 = S 2 − S 1, a 3 = S 3 − S 2

and

a n = S n − S n − 1

S n − 1 = n − 1 4 n − 1 + 1 = n − 1 4 n − 4 + 1 = n − 1 4 n − 3

And

S n = n 4 n + 1

.
Substituting values of

S n

and

S n − 1

a n = S n − S n − 1 = n 4 n + 1 − n − 1 4 n − 3 = 4 n 2 + n − 4 n 2 − 3 n − 4 n + 3 = 4 n 2 + n − 4 n 2 + 3 n + 4 n − 3

So,

a n = 8 n − 3

.
Now, let’s find terms of the series from

a n = 8 n − 3

a 1 = 81 − 3 = 8 − 3 = 5

a 2 = 82 − 3 = 16 − 3 = 13

a 3 = 83 − 3 = 24 − 3 = 21

a 4 = 84 − 3 = 32 − 3 = 29

So, the required A.P is 5, 13, 21, 29, …. .
Therefore, the correct option is 2.

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