How many values of m are possible?
For the following two (02) items: Let (6+10+14+\dots \text{up to } m \text{ terms}) = (1+3+5+7+\dots \text{up to } n \text{ terms}) where m \lt 25 and n \lt 25.
- A. None
- B. One
- C. Two ✓
- D. More than two
Correct Answer: C. Two
Explanation
We derived n^2 = 2m(m+2). For n^2 to be a perfect square, we test integer values for m \lt 25. If m=2, 2(2)(4) = 16 = 4^2 \implies n=4. If m=16, 2(16)(18) = 576 = 24^2 \implies n=24. Finding solutions amounts to solving a Pell-like equation, and the only integer solutions satisfying m \lt 25 and n \lt 25 are m=2 and m=16. Thus, there are two possible values.
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