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If the 12th term of an A.P. is - 13 and the sum of the first four terms is 24, what is the sum of the first 10 terms?

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answer is A.

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Detailed Solution

We are given that the

1 2^{th}

term of an A.P. is - 13 and the sum of the first four terms is 24.
We know,
The sum of the arithmetic progression is given as,

S n = n 2 2 a + n − 1 d .

where,
The number of terms is n.
The first term is a.
The common difference is d.
The general term of an A.P. is given by,

a n = a + (n − 1) d

According to the question,

a_{12} = a + 11 d

Again,

a 12 = a + (12 − 1) d ⇒ − 13 = a + 11 d ⇒ a = − 13 − 11 d

The sum of first four terms is given as,

24 = 42 2 − 13 − 11 d + 4 − 1 d ⇒ 24 = 2 − 26 − 22 d + 3 d ⇒ 24 = 2 − 26 − 19 d

⇒ − 26 − 19 d = 12 ⇒ 19 d = − 26 − 12 ⇒ 19 d = − 38

⇒ d = − 3819 ⇒ d = − 2

Also,

⇒ a = − 13 − 11 − 2 ⇒ a = − 13 + 22 ⇒ a = 9

Therefore, the sum of first 10 terms will be,

S 10 = 102 29 + 10 − 1 − 2 ⇒ S 10 = 5 18 − 18 ⇒ S 10 = 0

The sum of the first 10 terms is 0.
Hence, option 1 is correct.

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