Question

can anyone answer this pls​

Answer 1

Question :

Two AP.s have same common difference. the first term of one AP is 2 and that of the other is 7. the difference between their tenth terms is the same as the difference between their twenty first terms, which is same as the difference any two corresponding terms. Why ?

Theory :

General term of an AP

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

Solution :

Given : First 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

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

Hence , 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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More About Arithmetic Progression:

Sum of n terms of an AP given by :

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