Question

The sum and product of three consecutive terms of an AP is given to be 6 and -90. If the common difference is known to be negative, then the ${12}^{\text{th}}$ term of the AP is ___.

• A. -75
• B. 79
• C. -70
• D. -68

Answer 1

The correct option is A -75

Let the terms of the AP be $a-d,a$ and $a+d.$
By question, we have
$a-d+a+a+d=6.$
$\Rightarrow 3a=6$
$\Rightarrow a=2$
Also, by question, we have $(a-d)a(a+d)=-90.$
i.e., $(2-d)2(2+d)=-90$
$\Rightarrow 2(4-d^2)=-90$
$\Rightarrow 8-2d^2=-90$
$\Rightarrow 2d^2=98$
$\Rightarrow d^2=49$
$\Rightarrow d=\pm 7$
$d<0\Rightarrow d=-7$
The $n^{\text{th}}$ term of an AP of first term $a$ and common difference $d$ is given by
$a_n=a+(n-1)d.$
$\Rightarrow a_{12}=a+11d=2+11(-7)=-75$