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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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