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.P.s be

A.P.₁ = a₁, a₁ + d, a₁ + 2d, ...

A.P.₂ = a₂, a₂ + d, a₂ + 2d, ...

In A.P.₁ we have a₁ = 2

In A.P.₂ we have a₂ = 7

t₁₀ in A.P.₁ = a₁ + 9d = 2 + 9d   ...(1)

t₁₀ in A.P.₂ = a₂ + 9d = 7 + 9d   ...(2)

The difference between their 10^th terms

= (1) – (2) = 2 + 9d – 7 – 9d

= – 5   ...(I)

t₂₁ m A.P.₁ = a₁ + 20d = 2 + 20d   ...(3)

t₂₁ in A.P.₂ = 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.