Question

If $8$ times the $8th$ term of an arithmetic progression is equal to $12$ times the $12th$ term, then the $20th$ term is

• A. $12$ times to $8th$ term
• B. $8$ times to $12th$ term
• C. $0$
• D. Cannot be determined

Answer 1

Answer: (C)

$0$

Let $a$ be the first term and $d$ be their common difference of the AP.
Then, $nth$ term of the AP $T_n=a+(n-1)d$
Given, $8\times T_8=12\times T_{12}$
$\Rightarrow 8(a+(8-1\left)d\right)=12(a+(12-1\left)d\right)$
$\Rightarrow 8(a+7d)=12(a+11d)$
$\Rightarrow 2(a+7d)=3(a+11d)$
$\Rightarrow 2a+14d=3a+33d$
$\Rightarrow a=-19d$
So, $T_{20}=a+(20-1)d=-19d+19d=0$