Question

The first-term of an A.P. of non-constant terms is $3$ and it's second, tenth and thirty-fourth terms form a G.P., find the common difference.

Answer 1

nth term $t_n=a+(n-1)d$

$a=$ First term

$d=$ Common difference

$n=$ number of terms

$t_n=n^{th}$ term

$\therefore T_2=3+d,T_{10}=3+9d,T_{34}=3+33d$

Since $T_2,T_{10},T_{34}$ are in G.P

$\therefore T_{10}^2=T_2{T}_{34}$

$\Rightarrow (3+9d{)}^2=(3+d\left)\right(3+33d)$

$\Rightarrow d=0,1$

Here, d can not be zero as A.P. of non-constant terms.

Hence $d=1$