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Find the sum of first $25$ terms of an A.P. from the following options whose $n t h$ term is $1 − 4 n$ .

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1235

-1255

-1275

1275

answer is C.

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Detailed Solution

Given that,

n t h

term is

1 − 4 n

.
Take values for

n

from

1, 2, 3, ...

and substitute in

1 − 4 n

to formulate the sequence.
For

n = 1

a 1 = 1 − 41 ⇒ a 1 = − 3

Therefore the first term is

a = − 3

.
For

n = 2

a 2 = 1 − 42 ⇒ a 2 = − 7

Therefore the second term is,

a 2 = − 7

For

n = 3

a 3 = 1 − 43 ⇒ a 3 = − 11

Therefore the third term is,

a 3 = − 11

Thus the arithmetic progression is,

− 3, − 7, − 11, ...

.
Substitute

a = − 3, d = − 4, n = 25

in the equation

S n = n 2 2 a + n − 1 d

to calculate the sum.

S 25 = 252 2 − 3 + 25 − 1 − 4 ⇒ S 25 = 252 − 6 − 96 ⇒ S 25 = 252 − 102 ⇒ S 25 = − 1275

The sum of the 25 terms is -1275.
Hence, option 3) is correct.

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