In a certain A.P. the 24th term is twice the 10th term. Prove that t

1.	In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
Answer» Given, 24^th term is twice the 10^th term. We know that, n^th term a_n = a + (n – 1)d ⇒ a₂₄ = 2(a₁₀) a + (24 – 1)d = 2(a + (10 – 1)d) a + 23d = 2(a + 9d) a + 23d = 2a + 18d a = 5d …. (1) Now, the 72^nd term can be expressed as a₇₂ = a + (72 – 1)d = a + 71d = a + 5d + 66d = a + a + 66d [using (1)] = 2(a + 33d) = 2(a + (34 – 1)d) = 2(a₃₄) ⇒ a₇₂= 2(a₃₄) Hence, the 72^nd term is twice the 34^th term of the given A.P.

In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Answer»

Given,

24^th term is twice the 10^th term.

We know that, n^th term a_n = a + (n – 1)d

⇒ a₂₄ = 2(a₁₀)

a + (24 – 1)d = 2(a + (10 – 1)d)

a + 23d = 2(a + 9d)

a + 23d = 2a + 18d

a = 5d …. (1)

Now, the 72^nd term can be expressed as

a₇₂ = a + (72 – 1)d

= a + 71d

= a + 5d + 66d

= a + a + 66d [using (1)]

= 2(a + 33d)

= 2(a + (34 – 1)d)

= 2(a₃₄)

⇒ a₇₂= 2(a₃₄)

Hence, the 72^nd term is twice the 34^th term of the given A.P.