If the 3^rd and the 9^th terms of an A.P. are 4 and $-8$ respectively, which term of this A.P. is zero

Let first term and common difference of the A.P. be $\text{'}a\text{'}\text{\hspace{0.17em}and\hspace{0.17em}}\text{'}d\text{'}$ respectively

then $a_n=a+(n-1)d$

3^rd term $a_3=a+(3-1)d$

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

9^th term $a_9=a+(9-1)d$

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

$\left(2\right)-\left(1\right)$,

$a+8d=-8$

$a+2d=4$

 $\overline{6d=-12}$

$d=\frac{-12}{6}$

$d=-2$

$\begin{array}{l}\Rightarrow a+2d=4\\ \Rightarrow a-4=4\\ \Rightarrow a=4+4=8\end{array}$

Let the $\text{'}n\text{'}$th term of A.P. be zero, then

$\begin{array}{l}a+(n-1)d=0\\ \Rightarrow 8+(n-1)(-2)=0\\ \Rightarrow 8-2n+2=0\\ \Rightarrow 10-2n=0\\ \Rightarrow 2n=10\end{array}$

$\begin{array}{l}\Rightarrow n=\frac{10}{2}\\ \Rightarrow n=5\end{array}$

$\therefore$  Fifth term of A.P. is zero.