In a certain AP the 24th term is twice the 10th . . .

Question : In a certain AP the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Doubt by Zoha

Solution : 

Given :  a₂₄ = 2[a₁₀]
To Prove : a₇₂ = 2[a₃₄]
Proof : 

We know, 
a_n=a+(n-1)d

where 

a =first term 

d = common difference

a₂₄ = a+(24-1)d
a₂₄ = a+23d — (1)

similarly

a₁₀=a+9d — (2)

a₇₂=a+71d — (3)

a₃₄=a+33d — (4)

Now,

a₂₄ = 2[a₁₀] (Given)

a+23d = 2[a+9d] 
[From equation (1) and (2)]

a+23d=2a+18d
23d-18d=2a-a

5d=a — (5)

Now, 
a₇₂=a+71d [From equation (3)]
a₇₂= a-a+a+71d 
[Adding and subtracting a]
a₇₂=2a-a+71d

a₇₂=2a-(5d)+71d 
[From equation (5)]
a₇₂=2a+66d
a₇₂=2[a+33d]
a₇₂=2[a+(34-1)d]
a₇₂=2[a34]

[a_n=a+(n-1)d]
Hence Proved.