Question 1205174: A sequence {𝑎𝑛} is defined by 𝑎1 = 2, 𝑎2 = 3 and 𝑎𝑛+2 = 3𝑎𝑛+1 − 2𝑎𝑛 for all 𝑛 = 1, 2, 3, ….
Prove by mathematical induction that
𝑎𝑛 = 2^𝑛−1 + 1 for all 𝑛 = 1, 2, 3, …. Found 2 solutions by math_tutor2020, ikleyn:Answer by math_tutor2020(3817) (Show Source):
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I assume that an = 2^n-1+1 refers to
To make "n-1" the exponent, surround it in parenthesis.
You should write an = 2^(n-1)+1
Base case:
n = 1
and
and
and lastly
The base case is done.
Inductive step:
Assume that for some integer k such that k > 3.
The goal is to show based on that assumption.
A bit of scratch work off to the side
Now onto the inductive step
Reindexing (replace each n with k-1)
Substitution. Use the inductive assumption shown above.
This wraps up the inductive step and the induction proof is complete.
We have shown that the assumption leads to
Therefore, the recursive sequence
has the closed form
A spreadsheet can be used to generate a list of examples.
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