Question

If 9 times the $9^{th}$ term of an AP is equal to 13 times the ${13}^{th}$ term, then the ${22}^{nd}$ term of the AP is:

• A. $0$
• B. $22$
• C. $220$
• D. $198$

Answer 1

The correct option is A $0$

Let the first term and common difference of given AP be a and d, respectively.

Given

$9.a_9=13.a_{13}$

$\Rightarrow 9[a+(9-1\left)d\right]=13[a+(13-1\left)d\right]$

$\Rightarrow 9[a+8d]=13[a+12d]$

$\Rightarrow 9a+72d=13a+156d$

$\Rightarrow 4a+84d=0$

$\Rightarrow 4[a+21d]=0$

$\Rightarrow 4[a+(22-1)d=0$

$\Rightarrow a+(22-1)d=0$

$\Rightarrow a_{22}=0$