Question

Sum of three consecutive terms of an AP is 24 and product is 312. Find the middle term and common diffeerence of the AP.

• A. $6\text{and}\pm 3$
• B. $8\text{and}\pm 3$
• C. $6\text{and}\pm 5$
• D. $8\text{and}\pm 5$

Answer 1

The correct option is D $8\text{and}\pm 5$

Let the given terms be $(a-d),a$ and $(a+d),$ where $d$ is common difference.

Given sum of terms = 24

$\therefore (a-d)+a+(a+d)=24$

$\Rightarrow 3a=24⟹a=8$

So the terms are $(8-d),8$ and $(8+d)$.

Given product of terms = 312.

$\therefore (8-d)\times 8\times (8+d)=312$

$\Rightarrow (8-d)(8+d)\times 8=312$

$\Rightarrow 8^2-d^2=\frac{312}{8}=39$

$\Rightarrow 64-d^2=39$

$\Rightarrow d^2=25$

$\Rightarrow d=\pm 5$

So, the middle term of the AP is 8 and common difference is $\pm 5.$