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 . 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 . Why ?

Answer 1

First term of 1st A.P is 2.
First term of 2nd A.P is 7.
Consider the difference of their 10th terms,
$$\begin{gathered} a_{10}-a{\text{'}}_{10}=a+9d-a\text{'}-9d\text{'} \\ =a-a\text{'}+9d-9d\text{'} \\ =2-7+0\left[d=d\text{'}\right] \\ =-5 \\ {a}_{21}-a{\text{'}}_{21}=a+20d-a\text{'}-20d\text{'} \\ =a-a\text{'}+20d-20d\text{'} \\ =2-7+0\left[d=d\text{'}\right] \\ =-5 \\ Therefore,a_{10}-a{\text{'}}_{10}=a_{21}-a{\text{'}}_{21}. \end{gathered}$$
The difference between any two corresponding terms of A.P's is same as the difference between their terms.