Question

Find the middle term of the A.P. 213, 205, 197, ...., 37.

Answer 1

Let a be the first term and d be the common difference.

We know that, n^th term = a_n = a + (n − 1)d

It is given that a = 213, d = −8 and a_n = 37

According to the question,

⇒ 37 = 213 + (n − 1)(−8)
⇒ 37 = 213 − 8n + 8
⇒ 8n = 221 − 37
⇒ 8n = 184
⇒ n = 23 .... (1)

Therefore, total number of terms is 23.

Since, there are odd number of terms.
So, Middle term will be ${\left(\frac{23+1}{2}\right)}^{\mathrm{th}}$ term, i.e., the 12^th term.

∴ a₁₂ = 213 + (12 − 1)(−8)
= 213 − 88
= 125

Thus, the middle term of the A.P. 213, 205, 197, ...., 37 is 125.