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State True/False.

The middle term of the AP $10, 7, 4, …, (− 62)$ is -26.

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True

False

answer is A.

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Detailed Solution

Given A.P. is

10, 7, 4, … ., (− 62)

First term:

t 1 = 10

, second term:

t 2 = 7

. Applying formula and determining the value of common difference:

∴ d = t 2 − t 1 ⇒ d = 7 − 10 = − 3

Let

t n = − 62

and apply formula of

n t h

term of A.P.:

∴ t n = a + (n − 1) d ⇒ − 62 = 10 + (n − 1) × (− 3) ⇒ − 62 − 10 = − 3 n + 3

⇒ − 72 = − 3 n + 3 ⇒ − 3 n = − 72 − 3 ⇒ n = − 75 − 3 ⇒ n = 25

Therefore, there are 25 terms in the A.P.
As we know that, if n is odd, then middle term:

n + 1 2 t h term

Applying formula and determining middle term of A.P.:

= 25 + 1 2 ′ ′ term = 262 th term = 13 th term

Apply formula of

n t h

term of A.P. for 13th term:

∴ t 13 = 10 + (13 − 1) (− 3) ⇒ t 13 = 10 + 12 × − 3 = 10 − 36 = − 26

Therefore, the middle term of A.P. is -26.
Thus, the given statement is true.
Hence, option (1) is correct.

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