Question

Ten times the tenth term of an A.P. is equal to fifteen times, its fifteenth term. Find the twenty-fifth term of this A.P.

Answer 1

Let the first term of an A.P. be a and common difference be d.

We know that,

a_n = a + (n − 1)d

Given,

Ten times the tenth term of an A.P. is equal to fifteen times its fifteenth term.

⇒ 10a₁₀ = 15a₁₅

⇒ 10(a + 9d) = 15(a + 14d)

⇒ 10a + 90d = 15a + 210d

⇒ 10a − 15a + 90d − 210d = 0

⇒ −5a − 120d = 0

⇒ 5a + 120d = 0

⇒ 5(a + 24d) = 0

⇒ a + 24d = 0

⇒ a = −24d .....(1)

25th term:

⇒ a₂₅ = a + (25 − 1)d

⇒ a₂₅ = −24d + (24)d [From equation 1]

⇒ a₂₅ = 0

Hence, the twenty-fifth term of this A.P. = 0