If the $2^{nd}$ term of an A.P. is $13$ and the $5^{th}$ term is $25$, what is its $7^{th}$ term?
Given: The $2^{nd}$ term of an A.P. is $13$ and the $5^{th}$ term is $25$.
To do: To find $7^{th}$ term.
Solution:
Let $a$ be the first term and $d$ be the common difference of the A.P.
As known $a_n=a+( n-1)d$
$\Rightarrow 2^{nd}$ term $a_2=a+( 2-1)d$
$\Rightarrow 13=a+d\ ........\ ( i)$
Similarly, $5^{th}$ term $a_5=a+( 5-1)d$
$25=a+4d\ ........\ ( ii)$
On substracting $( i)$ from $( ii)$
$a+4d-a-d=25-13$
$\Rightarrow 3d=12$
$\Rightarrow d=\frac{12}{3}$
$\Rightarrow d=4$
On substituting $d=4$ in $( i)$
$a+4=13$
$\Rightarrow a=13-4=9$
Therefore, $7^{th}$ term of the A.P. , $a_7=9+( 7-1)4$
$=9+24$
$=33$
Thus, $7^{th}$ term of the A.P. is $33$.
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