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

Given:

Given A.P. is $213, 205, 197, …, 37$.

To do:

We have to find the middle term of the given A.P.

Solution:

$a_1=a=213, a_2=205, l=37$

Common difference $d=205-213=-8$

Let there be $n$ terms in the given A.P.

This implies,

$l=a_n=213+(n-1)(-8)$

$37=213+n(-8)-1(-8)$

$37-213=-8n+8$

$8n=8+176$

$8n=184$

$n=\frac{184}{8}$

$n=23$

Here, $n=23$ is odd.

Therefore, $(\frac{n+1}{2})$th term is the middle term.

$\frac{n+1}{2}=\frac{23+1}{2}=\frac{24}{2}=12$

Middle term $a_{12}=213+(12-1)(-8)$

$=213+11(-8)$

$=213-88$

$=125$

The middle term of the given A.P. is $125$.

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Updated on: 10-Oct-2022

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