If d is the common difference of A, and D is the common difference of B, then which one of the following is <strong>ALWAYS</strong> correct?
Consider the following for the next three (03) items that follow : Let P be the sum of first n positive terms of an increasing arithmetic progression A. Let Q be the sum of first n positive terms of another increasing arithmetic progression B. Let P:Q=(5n+4):(9n+6)
- A. D \gt d ✓
- B. D \lt d
- C. 7D \gt 12d
- D. None of the above
Correct Answer: A. D \gt d
Explanation
The sum of n terms of an AP is a quadratic polynomial in n where the coefficient of n^2 is half the common difference. Since the ratio of sums is (5n+4)/(9n+6), we have d/2 \propto 5 and D/2 \propto 9. This gives \frac{d}{D} = \frac{5}{9}. For increasing APs, d \gt 0 and D \gt 0, so D = \frac{9}{5}d \implies D \gt d.
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