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# Find the eleventh term from the last term of the A.P. $27,\ 23,\ 19,\ ....,-65$

**Given:**AP: $27,\ 23,\ 19,\ ...,\ -65$.

**To do:**To find the eleventh term from the last term of the AP.

**Solution:**

AP is $27,\ 23,\ 19,\ .....,\ 65$

Here $a = 27,\ d =23-27=-4$

First we should find which term is $-65$,

Let $a_n=-65$

$\Rightarrow a+( n-1)d=-65$

On putting values

$\Rightarrow 27+( n-1)( -4)=-65$

$\Rightarrow 27+n( -4) -1( -4) =-65$

$\Rightarrow 27-4n + 4 =-65$

$\Rightarrow 31-4n =-65$

$\Rightarrow -4n = -65 -31$

$\Rightarrow -4n =-96$

$\Rightarrow n = \frac{96}{4}$

$\Rightarrow n = 24$

$11^{th}$ from last term =$( 24 -10)^{th}$ term=$14^{th}$ term from the first

We need to find out $a_{14}$

$a_n=a+( n-1)d$

On putting the values

$\Rightarrow a_{14}=27+( 14-1)( -4)$

$\Rightarrow a_{14}=27+( 13)( -4)=27-52=-25$

$\therefore 11^{th}$ term from the last $=$ $14^{th}$ term from the first $=25$.

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