Answer to Question #284278 in Discrete Mathematics for Jasmine

Question #284278

Use the Principle of Mathematical Induction to prove that (2𝑛)! /(2n)𝑛! is odd for all positive integers.


1
Expert's answer
2022-01-03T16:30:35-0500

For n=1, we have

(2n)!2!n!=2!2!!=1 is odd \frac{(2 n) !}{2 ! n !}=\frac{2 !}{2 ! !}=1 \text { is odd }

Hence the result is true for n=1 (base case)

Let it be true for n=K.

Now for n=k+1

(2k+2)!2k+1β‹…(k+1)!=(2k+2)(2k+1)(2x)!2β‹…2kβ‹…(k+1)β‹…k!=(2k)!2kβ‹…k!β‹…2(k+1)β‹…(2k+1)2(k+1)=(2k)!2kβ‹…k!β‹…(2k+1)where (2k)!2kβ‹…k!,(2k+1) are odd\begin{aligned} \frac{(2 k+2) !}{2^{k+1} \cdot(k+1) !} &=\frac{(2 k+2)(2 k+1)(2 x) !}{2 \cdot 2^{k} \cdot(k+1) \cdot k !} \\ &=\frac{(2 k) !}{2^{k} \cdot k !} \cdot \frac{2(k+1) \cdot(2 k+1)}{2(k+1)} \\ &=\frac{(2 k) !}{2^{k} \cdot k !} \cdot(2 k+1) \\ &\text{where}\ \frac{(2 k) !}{2^{k} \cdot k !} ,(2 k+1)\ \text{are odd} \end{aligned}

Hence the result is true for n=k+1.

Thus the result is true for all +ve integers by mathematical induction.


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