# Arithmetic and Geometric Progressions

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## AP: Sum up to 'n' Terms

Sum of first 25 terms in AP is 525, sum of the next 25 terms is 725, what is the common difference?
1. ${8 \over 25}$
2. ${4 \over 25}$
3. ${6 \over 25}$
4. ${1 \over 25}$

• Correct Answer
Choice A. ${8 \over 25}$

## Detailed Solution

The sum of the first 25 terms, S25 = 525
The sum of the next 25 terms, K25 = sum of next 25 terms = 725
a26 = a1 + 25d
a27 = a2 + 25d

K25 = a26 + a27 + ...... + a50
Or K25 = a1 + 25d + a2 + 25d + ...... + a25 + 25d
Or K25 = a1 + a2 + ...... + a25 + 25(25d)

Or K25 = S25 + 25(25d)
i.e., 725 = 525 + 25 x 25d
200 = 25 x 25d
8 = 25d
d = ${8 \over 25}$

Correct Answer: ${8 \over 25}$

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## More questions from Progressions

With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. Reinforce these ideas with these questions.