How Does Fibonacci Sequence Behave with Addition and Multiplication?

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Homework Statement



Proof by induction of the following: f(m+k) = f(m-1)* f(k) + f(m) * f(k+1)

Homework Equations



f(k+1) = f(k) + f(k-1)

The Attempt at a Solution



The only way I can figure to get multiplication from addition was to square both sides, but then I really get out of my league.

What is the relationship if you double a fib. #?
 
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Do you know how to get induction started when you have two variables?
 
No i think that is my first issue here
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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