Question

If the $3^{rd}$ and the $9^{th}$ terms of an AP are $4$ and $-8$ respectively, which term of this AP is zero?

Answer 1

Let $a$ be the $1^{st}$ term and $d$ be the common difference.

$T\left(n\right)=a+(n-1)d$

Now, as per given condition,

$4=a+(3-1)d$

$4=a+2d.$..........................................$.\left(1\right)$

$-8=a+(9-1)d$

$-8=a+8d$........................................$.\left(2\right)$

On solving $a+8d-a-2d=-8-4=-12$

$\Rightarrow 6d=-12$ or $d=\frac{-12}{6}=-2$

Put $d=-2$ in $\left(1\right)$ we get

$4=a+2\times -2$

$\Rightarrow a=4+4=8$

Now,$0=a+(n-1)d$

$0=8+(n-1)(-2)$

$2n=10$

$\therefore n=5$

$\therefore 5^{th}$ term is zero.