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Solve (manually) for $a$:

$38 = \large \frac{(1+a)^{48} - 1}{a*(1+a)^{48}}$


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This is equivalent to a degree $48$ polynomial equation. There is no general "formula" for solving this kind of equation. Various approximation techniques work well enough for practical purposes. – André Nicolas Feb 7 '13 at 16:20
@AndréNicolas I think you meant $49$. – 1015 Feb 7 '13 at 16:36
Yes, am looking at tiny screen. – André Nicolas Feb 7 '13 at 16:45
up vote 3 down vote accepted

To work by calculator write it as $a=\frac 1{38} \left(1- \frac 1{(1+a)^{48}}\right)$. Now you can iterate: start by pretending $(1+a)^{48}$ is pretty big, so our first approximation is $a=\frac 1{38}\approx 0.026$, then plug that into the right and continue. It doesn't converge very quickly. Wolfram Alpha gives a numeric answer of about $0.00996937$, but it takes over $20$ iterations to get below $0.01$. A spreadsheet makes it easy.

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It always helps to start in the right place. I started a Newton-Raphson computation at $1/76$ (halfway between 0 and 1/38) and found $0.009969365412$ in just six iterations. But it's far from manual. – Peter Phipps Feb 7 '13 at 17:07
@PeterPhipps: it later occurred to me that bisection would get there quickly, and that could be done manually. As I was working with a spreadsheet and had copy down to do the iteration, this was the fastest route to a solution. – Ross Millikan Feb 7 '13 at 17:15

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