# Induction Proof: Am I on the right track?

• elizaburlap
In summary, the conversation discusses using induction to prove a formula for a sequence given initial values. The attempt at a solution involves checking the formula for the first few terms and then using the inductive hypothesis to prove it for all values of n. The final step uses algebraic manipulation to show that the formula holds for n+1.
elizaburlap

## Homework Statement

Let a(1)=a(2)=5 and a(n+1)=a(n)+6a(n-1), n≥2
Use induction to prove that a(n)=(3^n)-(-2)^n for n≥1

Not applicable

## The Attempt at a Solution

I have check that a(3) = 5+6·5 = 35 = 3^3-(-2)^3 so it holds for n = 3.
The cases n = 1 and n = 2 are similar and also hold

So I assumed that it holds for n and considered

a(n+1) = a(n)+6a(n-1)
= (3^n-(-2)^n)+6(3^(n-1)-(-2)^(n-1))

The second term [6(3^(n-1)-(-2)^(n-1))] equals [2·3^n+3·(-2)^n].

So,

a(n+1) = (3^n-(-2)^n)+(2·3^n+3·(-2)^n)
= 3^(n+1)-(-2)^(n+1)

Is this a valid induction proof? Am I on the right lines here?

Thanks!

welcome to pf!

hi elizaburlap! welcome to pf!

(try using the X2 and X2 buttons just above the Reply box )
elizaburlap said:
So I assumed that it holds for n and considered

a(n+1) = a(n)+6a(n-1)
= (3^n-(-2)^n)+6(3^(n-1)-(-2)^(n-1)) …

yes, that's all (difficult to read , but) fine!

Thank you! This was my first attempt at an induction proof, so I wasn't too sure.

Oh! I see the x2 now, thanks :)

Math1115. (:p)

## 1. What is induction proof?

Induction proof is a mathematical method used to prove a statement or theorem for all natural numbers. It involves proving a base case, and then using the fact that if the statement is true for a certain number, it is also true for the next number.

## 2. How do I know if I am on the right track with an induction proof?

To know if you are on the right track, you should check if you have correctly stated the base case and the inductive hypothesis. You should also make sure that your inductive step is logically sound and follows the correct format.

## 3. What are the common mistakes to avoid in an induction proof?

Some common mistakes to avoid in an induction proof include assuming that the statement is true for all numbers without proving it for the base case, using an incorrect inductive step, and not considering all cases when using the inductive hypothesis.

## 4. How do I write an effective induction proof?

To write an effective induction proof, you should clearly state the base case, the inductive hypothesis, and the inductive step. You should also use precise and concise language and provide logical reasoning for each step.

## 5. Is induction proof the only way to prove a statement for all natural numbers?

No, there are other methods such as direct proof and proof by contradiction that can also be used to prove a statement for all natural numbers. However, induction proof is a commonly used and efficient method for such proofs.

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