Question

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

Hint:

Use simultaneous equations to find \( a \) and \( d \) in an arithmetic sequence.

Answer 1

Using the formula for the \( n \)th term:
\[ a_n = a + (n-1)d. \] For 3rd term: \[ a + 2d = 4. \] For 9th term:
\[ a + 8d = -8. \] Subtracting:
\[ ( a + 8d) - (a + 2d) = -8 - 4. \] \[ 6d = -12. \] \[ d = -2. \] Substituting in \( a + 2d = 4 \):
\[ a + 2(-2) = 4. \] \[ a - 4 = 4. \] \[ a = 8. \] Finding \( n \) where \( a_n = 0 \):
\[ 8 + (n-1)(-2) = 0. \] \[ 8 - 2(n-1) = 0. \] \[ 8 = 2(n-1). \] \[ n-1 = 4. \] \[ n = 5. \] Thus, the 5th term is zero.