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The 7th term of an AP is -4 and its 13th term is -16. Then which of the following is the AP?

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8,6,4,2

9,7,5,3

11,8,6,1

7,10,8,2

answer is A.

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Detailed Solution

Given that, the 7th term of an AP is -4 and its 13th term is -16.
Let

t 1, t 2, t 3, … … …, t n

is an AP, then common difference

d = t 2 − t 1 = t 3 − t 2 = … … = t n − t n − 1

.
Since, general series of AP:

a, a + d, a + 2 d, a + 3 d, … …, a + (n − 1) d

Let first term a and common difference d.
Formula for the

n t h

term:

t n = a + n − 1 × d

Applying the formula of

n t h

term of AP and calculate 7th term:

∴ t 7 = a + (7 − 1) d ⇒ t 7 = a + 6 d ⇒ a + 6 d = − 4

⇒ a = − 4 − 6 d --------(1)

Applying the formula of

n t h

term of AP and calculate 13th term:

∴ t 13 = a + (13 − 1) d ⇒ t 13 = a + 12 d ⇒ a + 12 d = − 6 ------(2)

Put the value from equation (1) to equation (2):

∴ − 4 − 6 d + 12 d = − 16 ⇒ − 4 + 6 d = − 16 ⇒ 6 d = − 12 ⇒ d = − 2

Put the value of d in equation (1):

∴ a = − 4 − 6 × − 2 ⇒ a = − 4 + 12 = 8

Putting the value of a and d in the general series

∴ (8), [8 + (− 2)], [8 + 2 × (− 2)], [8 + 3 × (− 2)], … … ⇒ (8), (8 − 2), (8 − 4), (8 − 6), … … ⇒ (8), (6), (4), (2), …

Therefore, AP series is 8,6,4,2
Hence, option (1) is correct.

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