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

1.	The 4th term of an A.P. is zero. Prove that its 25th term is triple its 11th term.
Answer» Given: a₄ = 0 To Prove: a₂₅ = 3 × a₁₁ Now, a₄ = 0 ⇒ a + 3d = 0 ⇒ a = –3d We know that, a_n = a + (n – 1)d a₁₁ = –3d + (11 – 1)d [from (i)] a₁₁ = –3d + 10d a₁₁ = 7d …(ii) Now, a₂₅ = a + (25 – 1)d a₂₅ = –3d + 24d [from(i)] a₂₅ = 21d a₂₅ = 3 × 7d a₂₅ = 3 × a₁₁ [from(ii)] Hence Proved

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

Answer»

Given: a₄ = 0

To Prove: a₂₅ = 3 × a₁₁

Now, a₄ = 0

⇒ a + 3d = 0

⇒ a = –3d

We know that,

a_n = a + (n – 1)d

a₁₁ = –3d + (11 – 1)d [from (i)]

a₁₁ = –3d + 10d

a₁₁ = 7d …(ii)

Now,

a₂₅ = a + (25 – 1)d

a₂₅ = –3d + 24d [from(i)]

a₂₅ = 21d

a₂₅ = 3 × 7d

a₂₅ = 3 × a₁₁ [from(ii)]

Hence Proved