SOLUTION: 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

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): You can put this solution on YOUR website!

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.

Answer by ikleyn(52908)   (Show Source): You can put this solution on YOUR website!
.

For typical problems on the method of Mathematical Induction, see the lessons
    - Mathematical induction and arithmetic progressions
    - Mathematical induction and geometric progressions
    - Mathematical induction for sequences other than arithmetic or geometric
    - Proving inequalities by the method of Mathematical Induction
    - OVERVIEW of lessons on the Method of Mathematical induction
in this site.

Learn the subject from there.

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