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how to solve the following equation:

$$0.2948(1-(1+x)^{-5})=x$$

I know to satisfy this equation $x$ should be equal to 0.145 but how i can get there please help!

Hasan Saad
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moe
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    Logarithms probably aren't very helpful (this is just a rational equation). You would get a $6$th degree polynomial equation multiplying both sides by $(1 + x)^5$. I expect numerical solutions to be as good as it gets. – pjs36 Aug 16 '15 at 19:45
  • how to solve it with a numerical solution ? @pjs36 could you give me a hint please ? – moe Aug 16 '15 at 20:04

2 Answers2

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Newton's method is very efficient for this sort of equation, especially when you have a good initial estimate. It's easy to guess $x\approx 0.2$ to begin. So, using Newton's iteration $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)},$$with $$f(x)=0.2948\left(1-\frac1{(1+x)^5}\right)-x,\quad\quad f'(x)=\frac{1.474}{(1+x)^6}-1,$$and $x_0=0.2$, we can successively compute $x_1,x_2,$ and so on. Just two iterations will be enough to give good accuracy, and three is plenty.

John Bentin
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HINT: expanding all and factorizing we get $$-x \left(2500 x^5+11763 x^4+21315 x^3+17630 x^2+5130 x-1185\right)=0$$ this is an polynomial in degree six. One solution is $x=0$ for the other you will need a numerical method.

Dr. Sonnhard Graubner
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