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I would like to demonstrate a recursive formula that I have inferred. My background in mathematics is not quite as high as I hoped and although I have tried to apply the basic tricks that go with the recursive demonstrations, I haven't had much success.

The context is 1D wave propagation in a rod that can be of varying impedance: $n$ is the number of impedance changes and $i_k = I_{k-1}/I_{k}$ is an impedance ratio characterizing the $n$-th impedance change. The problem I am facing with is this one: I find the following expressions for $n=0,1,\dots$ and I want to generalise it to $n$ (done) and have a formal demonstration of it.

For $n=0$, $v_0 = \displaystyle\frac{1}{I_0} \left( 2\Lambda - R \right)$

For $n=1$, $v_1 = \displaystyle\frac{1}{I_1} \left(2 \displaystyle\frac{2}{1+i_1}\Lambda - R \right)$

For $n=2$, $v_2 = \displaystyle\frac{1}{I_2} \left(2 \displaystyle\frac{4}{(1+i_1)(1+i_2)}\Lambda - R \right)$

From there I infer that for $n$, $v_n = \displaystyle\frac{1}{I_n} \left( \displaystyle 2 \prod_{k=1}^{n} \frac{2^{n} }{(1+i_k)} \Lambda - R \right)$ but I haven't found a way to properly demonstrate it.

I thank you in advance for you help.

Henri
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This looks like a good candidate for mathematical induction. Arturo Magidin has a nice writeup in his answer to this question. You have done the base case. Now you assume the equation is correct for $n$ and from that prove that it is true for $n+1$. Once you have done that, equation 2 implies equation 3, which implies equation 4, etc.

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Ross Millikan
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