Class 10

Math

All topics

Arithmetic Progressions

The first term of $A.P$ is $8,n_{th}$ terms is $33$ and sum of first $n$ terms is $123$, then find $n$ and common difference $d$.

$n_{th}$ term $(a_{n})=33$

sum of $n$ terms $(S_{n})=123$

$∵n_{th}$ term $a_{n}=a+(n−1)d$

$⇒33=8+(n−1)d$

$⇒(n−1)d=33−8$

$⇒(n−1)d=25$...........(i)

Now, sum of $n$ terms

$S_{n}=2n [2a+(n−1)d]$

$⇒123=2n [2×8+25]$ [from equation (i)]

$⇒123=2n (16+25)$

$⇒123=2n ×41$

$⇒n=41123×2 $

$⇒n=6$

Put the value of $n$ in equation (i)

$⇒(6−1)d=25$

$⇒5d=25$

$⇒d=5$

Thus, $n=6$ and $d=5$