If 9 times the 9th term of an AP is equal to 13 times the 13th term, show that its 22nd term is 0.

Given : 9 × (9th term) = 13 × (13th term)
To prove: 22nd term is 0
9 × (a + 8d) = 13 × (a + 12d)
9a + 72d = 13a + 156d

– 4a = 84d
a = – 21d …..Equation 1
Also 22nd term is given by
a + 21d
Using equation 1 we get
– 21d + 21d = 0
Hence proved 22nd term is 0.