Q22 of 176 Page 8

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

Given: 10 × a₁₀ = 15 × a₁₅

To Prove: a₂₅ = 0

Now,

10 × (a + 9d) = 15 × (a + 14d)

⇒ 10a + 90d = 15a + 210d

⇒ 10a – 15a = 210d – 90d

⇒ –5a = 120d

⇒ a = –24d …(i)

Now,

a_n = a + (n – 1)d

a₂₅ = –24d + (25 – 1)d [from (i)]

a₂₅ = –24d + 24d

a₂₅ = 0

Hence Proved

Which term of the A.P. 3, 10, 17, 24, ... will be 84 more than its 13th term ?

The 4th term of an A.P. is zero. Prove that its 25th term is triple its 11th term.

If (m + 1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.

If t_n be the nth term of an A.P. such that find .