#### Two APs have the same common difference. The first term of one AP is 2 and that of the other is 7. The difference between their 10th terms is the same as the difference between their 21stterms, which is the same as the difference between any two corresponding terms. Why?

Solution.

It is given that the first term of an AP is 2 and of other is 7.

$\\{ Let}\\ a\textsubscript{1} = 2 b\textsubscript{1} = 7\\ Common difference of both A.P is same \\ A\textsubscript{n} is nth term for first A.P,b\textsubscript{n} is nth term of second A.P\\ As per the question\\ a\textsubscript{10} - b\textsubscript{10} = a\textsubscript{1}+ 9d - b\textsubscript{1} - 9d\\ = a\textsubscript{1}- b\textsubscript{1} = 2 - 7 = - 5 \\ a\textsubscript{21 }- b\textsubscript{21} = a\textsubscript{1} + 20d - b\textsubscript{1} - 20d\\ = a\textsubscript{1} - b\textsubscript{1 }= - 5 \\ Difference between first term = a\textsubscript{1} - b\textsubscript{1 }= - 5\\$

it is because when we find the difference between 10th and 21st term it is equal to the a1-b1  which is the difference between first terms.