Question:
The least upper bound of the set $\{\frac{3n+2}{2n+1}/n\in N\}$ where N is set of natural numbers is}
The least upper bound of the set $\{\frac{3n+2}{2n+1}/n\in N\}$ where N is set of natural numbers is}
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For decreasing sequences, the first term ($n=1$) is always the Supremum.
Updated On: Jul 3, 2026
- $\frac{3}{2}$
- $\frac{5}{2}$
- $\frac{5}{3}$
- $\frac{2}{3}$
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The Correct Option is C
Solution and Explanation
Step 1: Concept The least upper bound (Supremum) is the smallest value that is greater than or equal to all elements in the set.
Step 2: Meaning Let $a_n = \frac{3n+2}{2n+1}$. We test the first few terms: for $n=1, a_1 = \frac{5}{3} \approx 1.66$; for $n=2, a_2 = \frac{8}{5} = 1.6$; for $n=3, a_3 = \frac{11}{7} \approx 1.57$.
Step 3: Analysis The function $f(n) = \frac{3n+2}{2n+1}$ is a decreasing function because as $n$ increases, the terms get smaller and approach $1.5$.
Step 4: Conclusion Since the sequence is decreasing, the first term ($n=1$) is the largest value in the set. Thus, the least upper bound is $\frac{5}{3}$. Final Answer: (C)
Step 2: Meaning Let $a_n = \frac{3n+2}{2n+1}$. We test the first few terms: for $n=1, a_1 = \frac{5}{3} \approx 1.66$; for $n=2, a_2 = \frac{8}{5} = 1.6$; for $n=3, a_3 = \frac{11}{7} \approx 1.57$.
Step 3: Analysis The function $f(n) = \frac{3n+2}{2n+1}$ is a decreasing function because as $n$ increases, the terms get smaller and approach $1.5$.
Step 4: Conclusion Since the sequence is decreasing, the first term ($n=1$) is the largest value in the set. Thus, the least upper bound is $\frac{5}{3}$. Final Answer: (C)
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