Question

The sum of the third and seventh terms of an A. P. is $6$ and their product $8$, then common difference is

• A. $\pm 1$
• B. $\pm 2$
• C. $\pm \frac{1}{2}$
• D. $\pm \frac{1}{4}$

Answer 1

The correct option is C $\pm \frac{1}{2}$

$a_3+a_7=6$
$\Rightarrow a+2d+a+6d=6$
$\Rightarrow 2a+8d=6$
$\Rightarrow a+4d=3$
$\Rightarrow a-2b=3$
$\Rightarrow a=3-4d$
$\left(a_3\right)\left(a_7\right)=8$
$\Rightarrow (a+2d)(a+6d)=8$
$\Rightarrow [3-4d+2d][3-4d+6d]=8$
$\Rightarrow (3-2d)(3+2d)=8$
$\Rightarrow 9-4d^2=8$
$\Rightarrow 4d^2=1$
$\Rightarrow d^2=\frac{1}{4}$
$\Rightarrow d=\pm \frac{1}{2}$