If the 3rd and the 9th terms of an AP are 4 and -8 respectively, which term of this AP is zero?
5th term
Step 1: Use the nth term formula of an AP.
For an arithmetic progression (AP), the nth term is:
\[a_n = a + (n - 1)d\]
where \(a\) is the first term and \(d\) is the common difference.
Step 2: Use the given 3rd term.
The 3rd term is 4, so:
\[a_3 = a + 2d = 4 \quad ...(1)\]
Step 3: Use the given 9th term.
The 9th term is -8, so:
\[a_9 = a + 8d = -8 \quad ...(2)\]
Step 4: Solve for the common difference \(d\).
Subtract equation (1) from equation (2):
\[(a + 8d) - (a + 2d) = -8 - 4\]
\[a + 8d - a - 2d = -12\]
\[6d = -12\]
So,
\[d = \dfrac{-12}{6} = -2\]
Step 5: Find the first term \(a\).
Substitute \(d = -2\) into equation (1):
\[a + 2(-2) = 4\]
\[a - 4 = 4\]
\[a = 8\]
Step 6: Find which term is zero.
We want \(a_n = 0\):
\[a_n = a + (n - 1)d = 0\]
Substitute \(a = 8\) and \(d = -2\):
\[8 + (n - 1)(-2) = 0\]
\[8 - 2(n - 1) = 0\]
\[8 - 2n + 2 = 0\]
\[10 - 2n = 0\]
\[2n = 10 \Rightarrow n = 5\]
Conclusion: The 5th term of this AP is zero.