A simple arithmetic progression problem, where you are given the various constraints and you need to find a specific term of the progression

1. Progressions Q2

2. Qn: Arithmetic Progression
(a) 7 times the 13th term
(b) 0
(c) 13 times the 7th term
(d) 4 times the 4th term + 9 times the 9th term
If 4 times the 4th term of an A.P. is equal to 9 times the 9th term of
the A.P., what is 13 times the 13th term of this A.P.?

3. Soln: Arithmetic Progression
4t4 = 9t9, we need to find t13
4(a + 3d) = 9(a + 8d)
4a + 12d = 9a + 72d
 5a + 60d = 0
 a + 12d = 0
 t13 = 0
 13 × t13 = 0
As a simple rule, if n × tn = m × tm, then tm+n = 0. See, if you can prove this.
Answer choice (b)
If 4 times the 4th term of an A.P. is equal to 9 times the 9th term of
the A.P., what is 13 times the 13th term of this A.P.?