Class 10

Math

All topics

Sequences and Series

If the sum of $n$ terms of an A.P. is $3n_{2}+5n$ and its $m_{th}$ term

is $164,$ find the value of $m.$

Taking $n=1,$ we get

$S_{1}=3(1)_{2}+5(1)$

$⇒S_{1}=3+5$

$⇒S_{1}=8$

$⇒a_{1}=8$

Taking $n=2,$ we get

$S_{2}=3(2)_{2}+5(2)$

$⇒S_{2}=12+10$

$⇒S_{2}=22$

$∴a_{2}=S_{2}−S_{1}=22−8=14$

Taking $n=3,$ we get

$S_{3}=3(3)_{2}+5(3)$

$⇒S_{3}=27+15$

$⇒S_{3}=42$

$∴a_{3}=S_{3}−S_{2}=42−22=20$

So, $a=8,$

$d=a_{2}−a_{1}=14−8=6$

Now, we have to find the value of $m$

$a_{n}=a+(n−1)d$

$⇒a_{m}=8+(m−1)6$

$⇒a_{m}=8+6m−6$

$⇒164=2+6m$

$⇒162=6m$

$⇒m=27$