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State True/False.

The middle term of the AP 6, 13, 20,…….,216 is 111.

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True

False

answer is A.

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Detailed Solution

Given A.P. is

6, 13, 20, … ., 216

.
So, first term:

t 1 = 6

, second term:

t 2 = 13

∴ d = t 2 − t 1 ⇒ d = 13 − 6 = 7

Let 216 be the nth term of A.P. and apply the formula for the nth term of A.P.

∴ t n = a + (n − 1) d ⇒ 216 = 6 + (n − 1) × 7 ⇒ 216 − 6 = 7 n − 7

⇒ 210 = 7 n − 7 ⇒ n = 2177 ⇒ n = 31

Thus, there are 31 terms in A.P.
Since we know that, if n is odd then then middle term is,

n + 1 2 t h term

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

= 31 + 1 2 ′ ′ term = 322 th term = 16 th term

Applying formula of

n t h

term and determining 16th term:

∴ t 16 = 6 + (16 − 1) × 7 ⇒ t 16 = 6 + 15 × 7 = 6 + 105 = 111

Therefore, the middle term of A.P. is 111.
Thus, the given statement is true.
Hence, option (1) is correct.

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