Solving for n in the equation $\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$ 
Solving for $n$ in the equation $$\left ( \frac{1}{2} \right )^{n}+\left ( \frac{1}{4} \right )^{n}+\left ( \frac{3}{4} \right )^{n}=1$$

Can anyone show me a numerical method step-by-step to solve this?
Thanks

A non-standard approach to get an approximate solution: 
Notice that the sum $$\frac14\left(\frac120.00^n+0.25^n+0.50^n+0.75^n+1.00^n\frac12\right)$$ is the five-points trapezoidal approximation of
$$\int_0^1t^ndt=\frac1{n+1}.$$
So,
$$f(n)=0.25^n+0.50^n+0.75^n\approx\frac4{n+1}-\frac12.$$
Hence, $f(n)=1$ when
$$n\approx\frac53=1.66666\cdots.$$
Note that
$$f(\frac53)=1.033304\cdots,$$not so bad.
