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Among the following choices, the number of terms of an AP 3, 7, 11, 15,… that will make the sum 406 are

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answer is C.

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Detailed Solution

It is given that 3, 7, 11, 15,... is an AP.
The sum of the terms is 406.
We have to find the number of terms.
From the given information,

a = 3

.
The common difference is given by,

d = 7 − 3

d = 4

The sum of the terms is 406, so,

S n = 406

.
The formula which gives the sum of the formula is,

S n = n 2 [2 a + (n − 1) d]

.
Substitute the known values in the sum of n terms formula, we get,

406 = n 2 [2 × 3 + (n − 1) (4)]

406 × 2 = n [6 + 4 n − 4]

812 = 2 n + 4 n 2

2 n 2 + n − 406 = 0

Factorize the above equation,

2 n 2 + 29 n − 28 n − 406 = 0 n (2 n + 29) − 14 (2 n + 29) = 0 (2 n + 29) (n − 14) = 0

When,

2 n + 29 = 0 n = − 292

This is not possible since the number is negative.
When,

n − 14 = 0 n = 14

The 14 terms will make the sum of the terms as 406.
Hence, option 3) is correct.

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