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vitfil [10]
2 years ago
13

Prove for every positive integer n that 2! * 4! * 6! ... (2n)! ≥ [(n + 1)]^n.

Mathematics
1 answer:
MArishka [77]2 years ago
8 0

Answer:Given below

Step-by-step explanation:

Using mathematical induction

For n=1

2!=2^1

True for n=1

Assume it is true for n=k

2!\cdot 4!\cdot 6!\cdot 8!.......2k!\geq \left ( k+1\right )^{k}

For n=k+1

2!\cdot 4!\cdot 6!\cdot 8!.......2k!2\left ( k+1\right )!\geq \left ( k+1\right )^{k}\dot \left ( 2k+2\right )!

because value of 2!\cdot 4!\cdot 6!\cdot 8!.......2k!=\left ( k+1\right )^{k}

\geq \left ( k+1\right )^{k}\dot \left ( 2k+2\right )!

\geq \left ( k+1\right )^{k}\left [ 2\left ( k+1\right )\right ]!

\geq \left ( k+1\right )^{k}\left ( 2k+\right )!\left ( 2k+2\right )

\geq \left ( k+1\right )^{k+1}\left ( 2k+\right )!

Therefore 2!\cdot 4!\cdot 6!\cdot 8!.......2k!2\left ( k+1\right )! must be greater than \left ( k+1\right )^{k+1}

Hence it is true for n=k+1

2!\cdot 4!\cdot 6!\cdot 8!.......2k!2\left ( k+1\right )!\geq \left ( k+1\right )^{k+1}

Hence it is true for n=k

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