Question

The sum of the third and seventh terms of an A.P is $6$ and their product is $8$ 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 $\pm \frac{1}{2}$

Let a be the first term and d be the common difference of the given AP.
Given $a_3+a_7=6$
$\Rightarrow (a+2d)+(a+6d)=6$
$\Rightarrow 2a+8d=6$
$\Rightarrow a+4d=3\Rightarrow a=3-4d$ ...(1)
Again given $\left(a_3\right)\left(a_7\right)=8$
$\Rightarrow (a+2d)(a+6d)=8$
$\Rightarrow (3-4d+2d)(3-4d+6d)=8$
$\Rightarrow (3-2d)(3+2d)=8$
$\Rightarrow (9-4d^2)=8\Rightarrow 4d^2=1$
$\Rightarrow d^2=\frac{1}{4}\Rightarrow d=\pm \frac{1}{2}$
Hence option 'C' is correct.