Question

The fourth term of an AP is equal to three times the first term and the seventh term exceeds twice the third by one. Find the first term and the common difference

• A. $3,2$
• B. $5,4$
• C. $1,6$
• D. None of these

Answer 1

The correct option is A $3,2$

Let a,d and $a_n$ be the first term,common difference and $n^{th}$ term of given AP.
Given, $a_4=3a$
$\Rightarrow a+3d=3a$
$\Rightarrow 2a=3d$
$\Rightarrow a=\frac{3d}{2}$ ......(1)
$a_7=2a_3+1$ (given)
$\Rightarrow a+6d=2(a+2d)+1$
$\Rightarrow a+6d=2a+4d+1$
$\Rightarrow a-2d=-1$ .......(2)
Substitute a = $\frac{3d}{2}$ in equation (2)
$\frac{3d}{2}-2d=-1$
$\Rightarrow -d=-2$
$\Rightarrow d=2$
and $a$ = $\frac{3d}{2}$ = 3
So, first term is $3$ and common difference is $2$