solving the equation to find variables.

Question

I have an equation $M = P\left(\frac{J}{\left(1-\left(1+J\right)^{-N}\right)}\right)$

I am trying to solve it to find J but I am stuck at $\frac{P}{M} = \frac{\left(1 + J\right)^N - 1}{J(1+J)^N}$

can you please help me?

Thanks!

Answer 1

Hint. If one sets $$ X=1+J $$ then the given relation is $$ \frac{P}M=\frac{X^N-1}{X-1}\cdot \frac1{X^N} $$ that is $$ \frac{P}M=\left(X^{N-1}+X^{N-2}+\cdots+1 \right)\cdot \frac1{X^N} $$ or $$ P \cdot X^N-M\cdot\left(X^{N-1}+X^{N-2}+\cdots+1 \right)=0. $$ Finding $X$ in terms of parameters $M$ and $P$ is solving an equation of $N$-th degree which is handled numerically rather than symbolically.