Question

Question 4
Two AP’s have the same common difference. The first term of one AP 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

Let the same common difference of two AP’s is d.
Given that, the first term of first AP and second AP are 2 and 7 respectively, then the AP’s are

2, 2 + d, 2 + 2d, 2 + 3d,…….
And 7, 7 + d, 7 + 2d, 7 + 3d,……
Now, ${10}^{th}$ terms of first and second AP’s are 2 + 9d and 7 + 9d, respectively.
So, their differences is; 7 + 9d - (2 + 9d) = 5
Also, ${21}^{st}$ terms of first and second AP’s are 2 + 20d and 7 + 20d, respectively.
So, their difference is (7 + 20d) - (2 + 9d) = 5
Also, if $a_{n}and{b}_n$ are the nth terms of first and second AP.
Then, $b_n-a_n$ = [7 + (n – 1) d)] – [ 2 + (n – 1) d] = 5
Hence, the difference between any two corresponding terms of such AP’s is the same as the difference between their first terms.