CAT - questions and solution from progressions

Have given one interesting question from progressions. This is from challenge quiz I on our website. This  is not that difficult, but as with all good questions, there is a formula-based method of solving them and there is a more straightforward intuitive way of solving them. (This question can be solved without putting pen to paper. Try it)

Question

Sum of first 25 terms in AP is 525, sum of the next 25 terms is 725, what is the common difference?

1. 8/25
2. 4/25
3. 6/25
4. 1/5

Correct Answer : 8/25. Correct Choice : (1)

Explanatory Answer

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 = 8/25

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