# Help with complex (1 Viewer)

#### mathsbrain

##### Member

Very stuck on b) and c)...

#### Trebla

Since $\bg_white \alpha$ is a root then
$\bg_white \alpha^2+\alpha+1=0$
Which implies
$\bg_white \alpha^n+\alpha^{n-1}+\alpha^{n-2}=0$
Use the same argument for $\bg_white \beta$ and the result follows

For the strong induction step use the property that
$\bg_white T_{n+1}=-T_n-T_{n-1}$
and
$\bg_white \cos\left(\dfrac{2(n-1)\pi}{3}\right)=\cos\left(\dfrac{2n\pi}{3}-\dfrac{2\pi}{3}\right)$
and the result should follow

#### mathsbrain

##### Member
Since $\bg_white \alpha$ is a root then
$\bg_white \alpha^2+\alpha+1=0$
Which implies
$\bg_white \alpha^n+\alpha^{n-1}+\alpha^{n-2}=0$
Use the same argument for $\bg_white \beta$ and the result follows

For the strong induction step use the property that
$\bg_white T_{n+1}=-T_n-T_{n-1}$
and
$\bg_white \cos\left(\dfrac{2(n-1)\pi}{3}\right)=\cos\left(\dfrac{2n\pi}{3}-\dfrac{2\pi}{3}\right)$
and the result should follow
is this an HSC question?

#### Drdusk

##### π
Moderator
is this an HSC question?
You're the one who asked the question. Shouldn't you know lol

#### mathsbrain

##### Member
You're the one who asked the question. Shouldn't you know lol

#### Drdusk

##### π
Moderator
Ah that's not what I meant.

To answer your question I don't think it's a HSC question but I do recall seeing this in a Girra trial

#### mathsbrain

##### Member
also never quite understood difference between strong vs weak induction, any examples to help make sense of this?

#### fan96

##### 617 pages
also never quite understood difference between strong vs weak induction, any examples to help make sense of this?
Say you have a statement you're trying to prove by induction.

A normal induction proof would go like "if this statement is true for k then it must also be true for k + 1."

A strong induction proof would be more like "if this statement is true for all numbers less than or equal to k then it must also be true for k + 1."

This extra condition gives you more to work with, and you sometimes need it for harder proofs.

(You may not need to make the assumption for every single number less than k, but you would need to make the assumption for at least a few.)

I vaguely remember doing a question related to the Fibonacci numbers with strong induction - you can try to find it online.

#### CM_Tutor

##### Well-Known Member
I vaguely remember doing a question related to the Fibonacci numbers with strong induction - you can try to find it online.
One such question proves Binet's formula for the nth Fibonacci number as follows:

Define F0 = 0, F1 = 1, and Fk + 2 = Fk + 1 + Fk for non-negative integers k. Use strong induction to prove that

$\bg_white F_n = \frac{1}{\sqrt{5}} \bigg(\frac{1 + \sqrt{5}}{2}\bigg)^n - \frac{1}{\sqrt{5}} \bigg(\frac{1 - \sqrt{5}}{2}\bigg)^n$

for all non-negative integers n.

Strong induction is required as you need to assume a formula for Fk + 1 and Fk and so need to prove two consecutive values work in the first part.

An alternative proof (also requiring strong induction) is as follows:

Prove that provided x2 = x + 1, then $\bg_white x^n = xF_n + F_{n - 1}$ (with the above definition of Fn) for all integers n greater than 1. From this result, derive Binet's formula.

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