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Arithmetic Progressions — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsArithmetic Progressions3 Marks
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 $10^{\text {th }}$ terms is the same as the difference between their $21^{\text {st }}$ terms, which is the same as the difference between any two corresponding terms. Why?
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Answer

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

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