Question

In an $A.P$.the first term is $8$ and the $n^{th}$ terms is $33$. the sum fo first $n$ terms is $123$. Find $n$ and $d$.

Answer 1

In an $A.P,$

First term $a=8$,

$n^{th}$ term $a_n=33,$

The sum of first n terms $S_n=123$

i.e, $a+(n-1)d=33$

$⟹8+(n-1)d=33$

$⟹(n-1)d=25$......(i)

$S_n=123$

$⟹\frac{n}{2}[2a+(n-1\left)d\right]=123$

$⟹\frac{n}{2}[16+25]=123$

$⟹n=123\times \frac{2}{41}$

$\therefore n=6$

From (i) $⟹(n-1)d=25$

$⟹(6-1)d=25$

$⟹5d=25$

$\therefore d=5$

Answer is $n=6$,$d=5$