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

It is given that \(3^{rd}\) and \(9^{th}\) term of AP are 4 and –8 respectively.

It means \(a_3 = 4\) and \(a_9 = -8\)

Using formula \(a_n = a + (n - 1)d\), to find \(n^{th}\) term of arithmetic progression,

\(\Rightarrow \) 4 = a + (3 – 1)d and,
–8 = a + (9 – 1)d
\(\Rightarrow\) - 4 = a + 2d and,
-8 = a + 8d

These are equations in two variables.

Using equation 4 = a + 2d,
we can say that a = 4 - 2d

Putting value of a in other equation -8 = a + 8d,

\(\Rightarrow\) -8 = 4 - 2d + 8d
\(\Rightarrow\) -12 = 6d
\(\Rightarrow \) d = -2

Putting value of d in equation -8 = a + 8d,

\(\Rightarrow\) -8 = a + 8 (-2)
\(\Rightarrow \) -8 = a – 16
\(\Rightarrow \) a = 8

Therefore, first term = a = 8 and Common Difference = d = -2

We want to know which term is equal to zero.

Using formula \(a_n = a + (n - 1)d\) , to find \(n^{th}\) term of arithmetic progression,

\(\Rightarrow \) 0 = 8 + (n - 1) (-2)
\(\Rightarrow \) 0 = 8 - 2n + 2
\(\Rightarrow \) 0 = 10 - 2n
\(\Rightarrow \) 2n = 10
\(\Rightarrow \) n = 5

Therefore, \(5^{th}\) term is equal to 0.