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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 10th terms is the same as the difference between their 21st terms, which is the same as the difference between any two corresponding terms.
Solution
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 10th 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 21st terms is
(3) – (4)
= 2 + 20d – 7 – 20d
= – 5 ...(II)
I = II
Hence it is Proved.
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