Question

Two A.P’s have the same common difference. The first term of one A.P is 2 and that of the other is 7. Show that the difference between their 10^th terms is the same as the difference between their 21^st terms, which is the same as the difference between any two corresponding terms.

Answer 1

Let the two A.Ps be 

AP₁ = a₁, a₁ + d, a₁ + 2d,…

AP₂ = a₂, a₂ + d, a₂ + 2d,… 

In AP₁ we have a₁ = 2 

In AP₂ we have a₂ = 7 

t₁₀ in AP₁ = a₁ + 9d = 2 + 9d … (1) 

t₁₀ in AP₂ = a₂ + 9d = 7 + 9d … (2) 

The difference between their 10^th terms 

= (1) – (2) = 2 + 9d – 7 – 9d 

= -5 … (I) 

t₂₁ m AP₁ = a₁ + 20d = 2 + 20d … (3) 

t₂₁ in AP₂ = a₂ + 20d = 7 + 20d … (4) 

The difference between their 21^st terms is 

(3) – (4) 

= 2 + 20d – 7 – 20d = -5 … (II) 

I = II 

Hence it is Proved.