Question

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

Answer 1

In this AP $(213,205,197,....,37)$

First term, $a=213$

Common difference, $d=205-213=(-8)$

$T_n=37$..........(a)

Now, $T_n=a+(n-1)d$

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

$213-8n+8$

$T_n=221-8n$

using(a)

$37=221-8n$

$37-221=(-8n)$

$(-184)=(-8n)$

$184/8=n$

$n=23$

Hence this AP contains 23 terms..

Middle term=$\frac{n+1}{2}$ (for odd numbers)

Middle term=$\frac{23+1}{2}$

$=\frac{24}{2}$

$=12thterm$

so, the middle term of this AP is its 12th term..

12th term,

$T_{12}=a+(12-1)d$

$=213+11\times (-8)$

$=213-88$

$T_{12}=125$