Find the least positive integer

In summary, the least positive integer is the smallest whole number greater than zero and is often denoted as 1. To find the least positive integer in a set of numbers, one can arrange the numbers in ascending order or start with 1 and increment until the smallest positive integer is found. There can only be one least positive integer in a set of numbers, and it is often used in finding the greatest common divisor (GCD). In mathematical proofs and theories, the least positive integer serves as a base case and is used in defining concepts, sets, and various operations and algorithms.
  • #1
anemone
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Find the least positive integer $k$ such that $\displaystyle {2n\choose n}^{\small\dfrac{1}{n}}<k$ for all positive integers $n$.
 
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  • #2
[sp]Since \(\displaystyle {2n+2 \choose n+1} = \frac{(2n+2)(2n+1)}{(n+1)^2}{2n \choose n} = 4\frac{n+\frac12}{n+1}{2n\choose n} < 4{2n\choose n}\), and \(\displaystyle {2\choose 1} = 2 < 4\), it follows by induction that \(\displaystyle {2n\choose n} < 4^n\) and therefore \(\displaystyle {2n\choose n}^{\!\!1/n}<4.\)

On the other hand, \(\displaystyle {34\choose 17}^{\!\!1/17} >3.006 >3.\) So the least value of $k$ is $4$.[/sp]
 
  • #3
Thanks, Opalg for participating and your solution! Your method is a nice one, I enjoy reading it!

Solution by other:
Note that $\displaystyle {2n\choose n}<{2n\choose 0}+{2n\choose 1}+\cdots+{2n\choose 2n}=(1+1)^{2n}=4^n$

and for $n=5$, $\displaystyle {10\choose 5}=252>3^5$, we can conclude that $k=4$.
 
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FAQ: Find the least positive integer

What is the definition of "least positive integer"?

The least positive integer refers to the smallest whole number that is greater than zero. It is often denoted as 1 and is the starting point for counting in mathematics.

How do you find the least positive integer in a given set of numbers?

To find the least positive integer in a set of numbers, you can arrange the numbers in ascending order and then identify the smallest one. Alternatively, you can also start with the number 1 and check if it is included in the set; if not, keep incrementing until you reach the smallest positive integer.

Can there be more than one least positive integer in a set of numbers?

No, there can only be one least positive integer in a set of numbers. This is because the concept of "least" implies that it is the smallest or lowest value in the set.

What is the relationship between the least positive integer and the greatest common divisor (GCD)?

The least positive integer is often used in finding the GCD of two or more numbers. It is the smallest number that is a factor of all the given numbers, making it an important concept in finding the GCD.

How is the least positive integer used in mathematical proofs and theories?

The least positive integer is often used in mathematical proofs and theories as a starting point or base case. It is also used in defining mathematical concepts and sets, and plays a crucial role in various mathematical operations and algorithms.

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