How can I solve an equation in this form without using linear interpolation?
$$ \frac{1.0025^N - 1 }{1.005^N} = 0.4 $$
Usually I would have to guess two values that I know $N$ is between and then use the linear interpolation formula. But is there a way to solve it directly by hand without using linear interpolation? Thanks in advance!
If you let $a=\frac{1.0025}{1.005}$ and $b=\frac{1}{1.005}$, then your equation can be rewritten as $$a^N-b^N=0.4$$
Find the maximum of the left side by taking the derivative and solving for where it is zero: $$\frac{d}{dN}[a^N-b^N]= a^Nln(a)-b^Nln(b)=0$$ So $$a^N ln(a)=b^N ln(b)$$ $$(\frac{a}{b})^N=\frac{ ln(b) }{ ln(a) }$$
$$N ln(\frac{a}{b})=ln(\frac{ln(b)}{ln(a)})$$ $$N = \frac{ln( \frac{ln(b)}{ln(a)} )} {ln(\frac{a}{b})}= 278.1050$$
Plug that value into the left side of your formula and you will see that the maximum value there is 0.2504. The left side never reaches to 0.4.
So the formula has no real solution.