Two APs have the same common difference. The first term of one AP is 2 and that of the other is 7. Is it true that the difference between their 10th term is the same as the difference between their 21st terms?

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Detailed Solution

It is given that, common difference of two APs is the same and the first term is 2 and 7 respectively.
Let the two series be AP1 and AP2 .
Assuming the first term of AP1 and AP2 be a1 and A1, i.e.

a 1 = 2

and

A 1 = 7

.
And the last terms be

a n = a 1 + (n − 1) d

and

A n = A 1 + (n − 1) d

.
The common difference is d as they have the same common difference. .
According to the question,
Difference between their 10th terms,

7 + (10 − 1) d − [2 + (10 − 1) d]

⇒ 7 − 2 + 9 d − 9 d = 5

Difference between their 21st terms,

7 + (21 − 1) d − [2 + (21 − 1) d]

⇒ 7 − 2 + 20 d − 20 d = 5

As, the difference between their 10th term is the same as the difference between their 21st terms.
Therefore, the given statement is true.

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