Question

Find the middle term of the AP 213, 205, 197, … , 37.

Answer 1

Hint: In this question, general formulas related to an AP are required. The ${n^{th}}$ term of an AP with first term a and common difference d is-
${a_n} = a + \left( {n - 1} \right)d$

Complete step by step answer:
The given AP is 213, 205, .. 37
The first term a = 213
The common difference d = ${a_2} - {a_1}$ =205 - 213 = -8

Let the last term be the ${n^{th}}$term, which is 37. So, by applying the formula for ${n^{th}}$ term,
${a_n} = a + \left( {n - 1} \right)d$
37 = 213 + (n - 1)(-8)
8(n - 1) = 213 - 37 = 176
n - 1 = 22
n =23

We have to find the middle term of the AP. The total number of terms are 23, the middle term is the $\left(\dfrac{\mathrm n+1}2\right)^{\mathrm{th}}$ term.

So, the middle term is the ${12^{th}}$ term.
Again, by applying the formula for ${n^{th}}$ term-
${a_n}$= a + (n - 1)d
${a_n}$ = 213 + (12 - 1)(-8)
${a_n}$ = 213 - 88
${a_n}$ = 125

Hence, the middle term if the AP is 125.


Note: One should know how to find the middle term in the sequence. If the number of terms are even, then there are two middle terms and we have to write both of them. If the number of terms are even, then middle terms are-
$\left(\dfrac{\mathrm n}2\right)^{\mathrm{th}\;}\;\mathrm{and}\;\left(\dfrac{\mathrm n}2+1\right)^{\mathrm{th}}\;\mathrm{terms}$