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Chapter 4: Problem 11

If \(n \in \mathbf{Z}^{+}\)and \(n \geq 2\), prove that \(2^{n}<\left(\begin{array}{c}2 n \\ n\end{array}\right)<4^{n}\).

Short Answer

Expert verified
Proving the given inequality \(2^{n}<\left(\begin{array}{c}2 n \ n\end{array}\right)<4^{n}\) involves expanding the binomial coefficient and simplifying the factorial, then comparing the result with the powers of 2 and 4.

Step by step solution

01

Expand the Binomial Coefficient

The given binomial coefficient can be expanded as follows \(\left(\begin{array}{c}2 n \ n\end{array}\right) = \frac{(2n)!}{n!n!}\) This expression basically represents the number of ways \(2n\) elements can be chosen \(n\) at a time.
02

Simplify the Factorial

Next, we simplify the factorial expression. Factorial of a positive integer \(n\), written as \(n!\), is the product of all positive integers less than or equal to \(n\). Therefore, \( (2n)! = (2n)(2n-1)...(n+1)*n*n-1*...*1 \) . After simplification we get: \(\left(\begin{array}{c}2 n \ n\end{array}\right) = \frac{(2n)(2n-1)...(n+1)}{n!}\)
03

Compare with the Powers

Noting that \(2n > n\), \(2n -1 > n\) and so on up to \((n+1) > n\), multiply \(n\) factors greater than \(n\) to yield \(n^{n}\) and we see that \(\frac{(2n)(2n-1)...(n+1)}{n!} > n^{n}\), and since on exponentiation \(n^{n} < 2^{n}\), we obtain \( \frac{(2n)(2n-1)...(n+1)}{n!} > 2^{n}\) proving \( \left(\begin{array}{c}2 n \ n\end{array}\right) > 2^{n}\). Likewise, the expression \(\frac{(2n)(2n-1)...(n+1)}{n!} < (2n)^n = 2^{n} \cdot 2^{n} = 4^{n}\) thereby proving \(\left(\begin{array}{c}2 n \ n\end{array}\right) < 4^{n}\)

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Most popular questions from this chapter

a) Ten students enter a locker room that contains 10 lockers. The first student opens all the lockers. The second student changes the status (from closed to open, or vice versa) of every other locker, starting with the second locker. The third student then changes the status of every third locker, starting at the third locker. In general, for \(1

Let \(a, b \in \mathbf{Z}^{+}\)with \(a\) even and \(b\) odd. Prove that \(\operatorname{gcd}(a, b)=\operatorname{gcd}(a / 2, b)\).

Write a computer program (or develop an algorithm) to convert a positive integer in base 10 to base \(b\), where \(2 \leq b \leq 9\).

Let \(a, b \in \mathbf{Z}^{+}\). If \(b \mid a\) and \(b \mid(a+2)\), prove that \(b=1\) or \(b=2\).

a) Give a recursive definition for the intersection of the sets \(A_{1}, A_{2}, \ldots, A_{n}, A_{n+1} \subseteq\) \(\varkappa, n \geq 1\). b) Use the result in part (a) to show that for any \(n, r \in \mathbf{Z}^{+}\)with \(n \geq 3\) and \(1 \leq r
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