We have a recursively defined sequence a_{n}. a_{0}=0,a_{1}=3, and a_{n

Ava-May Nelson

Ava-May Nelson

Answered question


We have a recursively defined sequence an.
a0=0,a1=3, and an=3an12an2 for n2
We would like to prove that f or all n0,an=32n3.
Prove this using the stronger mathematical induction.

Answer & Explanation



Skilled2021-08-17Added 98 answers

Step 1
Given that: {an} be a sequance of defined recursively a0=0, a1=3
an=3an12an2 for n2
To show: an=32n3 for n2
we will use strong mathematical induction
for n=0 a0=3203=0
n=1 a1=3213=3
So, for n=0, n=1 it is true
Let us assume it is true for all kn
Such that ak=32k3 kn
Now for k=n+1
So, an+1=3an2an1
( by induction an=32n3, an1=32n13)

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