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Sum

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 10^{th} terms is the same as the difference between their 21^{st} terms, which is the same as the difference between any two corresponding terms. Why?

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#### Solution

Suppose there are two AP’s with first terms a and A

And their common differences are d and D respectively

Suppose n be any term

a_{n} = a + (n – 1)d

A_{n} = A + (n – 1)D

As common difference is equal for both AP’s

We have D = d

Using this we have

A_{n} – a_{n} = a + (n – – 1)d – [ A + (n – 1)D]

= a + (n – 1)d – A – (n – 1)d

= a – A

As a – A is a constant value

Therefore, difference between any corresponding terms will be equal to a – A.

Concept: Arithmetic Progression

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