If 9 times the 9^th term of an A.P. is equal to 13 times the 13^th term, then the 22^nd term of the A.P. is

We know that,

a_n = a + (n – 1)d

So, a₉ = a + (9 – 1)d

⇒ a₉ = a + 8d

and a₁₃ = a + 12d

According to the question,

9 times the 9^th term i.e. a₉ = 13 times the 13^th term i.e. a₁₃

⇒ 9 × a₉ = 13 × a₁₃

⇒ 9(a + 8d) = 13(a + 12d)

⇒ 9a + 72d = 13a + 156d

⇒ 9a – 13a = 156d – 72d

⇒ -4a = 84d

⇒ a = -21d …(i)

Now, we have to find the 22^nd term

∴ a₂₂ = a + 21d

⇒ a₂₂ = -21d + 21d [from (i)]

⇒ a₂₂ = 0

Hence, the correct option is (a)