In an AP, the ratio of the sum of the first p terms to the sum of the first q terms is p^{2}:q^{2}. Which one of the following is correct?
- A. The first term is equal to the common difference
- B. The first term is equal to twice the common difference
- C. The common difference is equal to twice the first term ✓
- D. The first term is equal to square of the common difference
Correct Answer: C. The common difference is equal to twice the first term
Explanation
Given \frac{\frac{p}{2}[2a+(p-1)d]}{\frac{q}{2}[2a+(q-1)d]} = \frac{p^2}{q^2}. Cancelling out common terms gives \frac{2a+(p-1)d}{2a+(q-1)d} = \frac{p}{q}. Cross-multiplying yields 2aq + pqd - qd = 2ap + pqd - pd, which simplifies to 2a(q-p) = d(q-p). Assuming p \neq q, we deduce that d = 2a.
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