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8090 [49]
3 years ago
11

Prove that sequence an = (3^n)/(2n)! is monotone

Mathematics
1 answer:
krek1111 [17]3 years ago
5 0
When n=1, you have a_1=\dfrac3{2!}=\dfrac32.

When n=2, you have a_2=\dfrac{3^2}{4!}=\dfrac38.

Clearly, a_1>a_2.

Assume a_k. Now when n=k+1, you have

a_{k+1}=\dfrac{3^{k+1}}{(2k+2)!}=\dfrac3{(2k+2)(2k+1)}\times\dfrac{3^k}{(2k)!}=\dfrac3{(2k+2)(2k+1)}a_k

Therefore by induction the sequence is monotone (decreasing).
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An aquarium,which is a right rectangular prism.

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