If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that the 25th term of the A.P. is Zero.

Answer:

Given,

10 times the 10^th term of an A.P. is equal to 15 times the 15^th term

10a₁₀ = 15a₁₅

a_n = a + (n – 1) d

When n = 10,

a₁₀ = a + (10 – 1)d

= a + 9d

When n = 15,

a₁₅ = a + (15 – 1)d

= a + 14d

When n = 25,

a₂₅ = a + (25 – 1)d

= a + 24d ………(i)

 

10a₁₀ = 15a₁₅

10(a + 9d) = 15(a + 14d)

10a + 90d = 15a + 210d

10a – 15a + 90d – 210d = 0

-5a – 120d = 0

-5(a + 24d) = 0

a + 24d = 0

a₂₅ = 0

Thus, Proved.