Question

Find the middle term of the AP 6, 13, 20, ..., 216.

Answer 1

The given AP is 6, 13, 20, .., 216.

First term, a = 6

Common difference, d = 13 – 6 = 7

Suppose these are n terms in the given AP. Then,

a_n = 216

⇒ 6 + (n – 1) × 7 = 216   ...[a_n = a + (n – 1)d]

⇒ 7(n – 1) = 216 – 6 = 210 

⇒ `n - 1 = 210/7 = 30`

⇒ n = 30 + 1 = 31 

Thus, the given AP contains 31 terms,

∴ Middle term of the given AP

= `((31 + 1)/2)`th term

= 16th term

= 6 + (16 – 1) × 7 

= 6 + 105

= 111

Hence, the middle term of the given AP is 111.