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.

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