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The sum of the first 10 terms of an AP is -150 and the sum of its next 10 terms is -550. What is the AP?

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4, -1, -7, -8,.....

1, -1, -4, -7,.....

3, -1, -5, -9,.....

2, -3, -6, -9,.....

answer is C.

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Detailed Solution

Given:

S 10 = − 150

and the sum of its next 10 terms is −550.
Let a be the first term and d be the common difference of the A.P.
Formula for sum of n terms of an A.P. formula:

S n = n 2 [2 a + (n − 1) d]

Applying sum of nth term of an A.P. formula in first given condition:

∴ S 10 = − 150 ⇒ 102 (2 a + 9 d) = − 150 ⇒ 5 (2 a + 9 d) = − 150 ⇒ 2 a + 9 d = − 30 − − − − − − (1)

∴ S 20

= sum of first 20 terms.

⇒ S 20

= sum of first 10 terms + sum of next 10 terms.

= − 150 + (− 550) = − 700

Applying sum of nth term of an A.P. formula in second given condition:

∴ S 20 = − 700 ⇒ 202 (2 a + 19 d) = − 700 ⇒ 10 (2 a + 19 d) = − 700 ⇒ 2 a + 19 d = − 70 − − − − − (2)

Subtract equation (1) from equation (2):

∴ 2 a + 19 d − 2 a − 9 d = − 70 − (− 30) ⇒ 10 d = − 40 ⇒ d = − 40 10 ⇒ d = − 4

Substitute the value of d in equation (1):

∴ 2 a + 9 × (− 4) = − 30 ⇒ 2 a = − 30 + 36 ⇒ 2 a = 6 ⇒ a = 3

Therefore, the required A.P. is 3, -1, -5, -9,.....
Hence, option (3) is correct.

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