If 9 times the 9th term of an A.P. is equal to 13 times the 13th ter

1.	If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. isA. 0B. 22C. 220D. 198
Answer» 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)

If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. isA. 0B. 22C. 220D. 198

Answer»

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)