Question

two APS have the same common difference . The first term of an ap is 2 and that of the other is 7 . The difference between their tenth terms is the same as the difference between their 21st terms , which is the same as the difference between any two corresponding terms . why ?​

Answer 1

Answer:

Answer: The difference between consecutive terms in both APs is same. That is the reason, all corresponding terms in both APs have same difference. difference in consecutive terms in second AP = y

Step-by-step explanation:

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Answer 2

Solution :

Given : First term of first AP is 2 and first term of second AP is 7.

Let both Ap's have same common difference = d

General term of first Ap

$\sf \: a_{n} = 2 + (n - 1)d$

and General term of second Ap

$\sf \: a_{n} = 7 + (n - 1)d$

Accordingly to the question:

The difference between their tenth terms is the same as the difference between their twenty first terms .

Now 10th term of both the Ap.s

10 th term of first Ap = 2+9d

and 10 th term of second Ap = 7+ 9 d

$\implies \sf \: Diffrence \: = 7 + 9d - 2 - 9d = 5$

Also , 21 st term of both the Ap.s

21st term of first Ap = 2+20d

21 st term of second Ap= 7+20d

$\sf \implies\: Difference=7 + 20d - 2 - 20d = 5$

Also , Difference between their nth terms

$\sf \implies \: diffrence \: = 7 + (n - 1)d - (2 + (n - 1)d) = 5$

Therefore , the difference between any two corresponding terms is constant i.e 5.Because ,it does not depends on n & d as common difference,d is equal.

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Theory :

General term of an AP

$\sf \: a_{n} = a + (n - 1)d$

More About Arithmetic Progression:

Sum of n terms of an AP given by :

$\sf \: S_{n} = \dfrac{1}{2}(2a+ (n - 1)d)$