Que: If n is an odd integer, which of the following MUST BE an integer?
I. \(\frac{n^2-1}{2}\)
II. \(\frac{n-1}{2}\)
III. \(\frac{n}{2}\)
A. I only
B. II only
C. III only
D. I & II only
E. I, II, & III
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Que: If n is an odd integer, which of the following MUST BE an integer?
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Max@Math Revolution
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A
B
C
D
E
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Max@Math Revolution
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Solution: Multi-Choice questions: ‘Must be’ VS ‘Could be’.
We have to find the options that must be true
‘MUST BE’: Choose only the options that are absolutely true in ALL cases
=> ‘n’ is an odd integer, Always an integer
=> n = odd = 2m+1 (m is always an integer)
I. \(\frac{n^2-1}{2}=\frac{\left[\left(2m+1\right)^2-1\right]}{2}=\frac{\left[4m^2+4m+1-1\right]}{2}=2m^2+m\) => AN INTEGER ∴ Option is TRUE
II. \(\frac{n-1}{2}=\frac{\left[\left(2m+1\right)-1\right]}{2}=m\) => AN INTEGER ∴ Option is TRUE
III. \(\frac{n}{2}=\frac{\left(2m+1\right)}{2}=m+0.5\) => NOT AN INTEGER ∴ Option is NOT TRUE
Only options I and II are TRUE.
Therefore, D is the correct answer.
Answer D
We have to find the options that must be true
‘MUST BE’: Choose only the options that are absolutely true in ALL cases
=> ‘n’ is an odd integer, Always an integer
=> n = odd = 2m+1 (m is always an integer)
I. \(\frac{n^2-1}{2}=\frac{\left[\left(2m+1\right)^2-1\right]}{2}=\frac{\left[4m^2+4m+1-1\right]}{2}=2m^2+m\) => AN INTEGER ∴ Option is TRUE
II. \(\frac{n-1}{2}=\frac{\left[\left(2m+1\right)-1\right]}{2}=m\) => AN INTEGER ∴ Option is TRUE
III. \(\frac{n}{2}=\frac{\left(2m+1\right)}{2}=m+0.5\) => NOT AN INTEGER ∴ Option is NOT TRUE
Only options I and II are TRUE.
Therefore, D is the correct answer.
Answer D
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