What is the relation between m and n?
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. n^{2}=m(m+1)
- B. n^{2}=m(m+2)
- C. n^{2}=2m(m+1)
- D. n^{2}=2m(m+2) ✓
Correct Answer: D. n^{2}=2m(m+2)
Explanation
The LHS is an AP with first term a=6 and common difference d=4. Its sum is \frac{m}{2}[2(6) + (m-1)4] = \frac{m}{2}[4m+8] = 2m(m+2). The RHS is the sum of the first n odd natural numbers, which is known to be n^2. Equating them gives n^2 = 2m(m+2).
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