If it makes any sense, what is the best way to find the root of a binomial expansion for any values of n, without using interpolation?
For example, how will you solve for a $${(1+a)^{25}}=\frac{3000a-300}{2500a-300}$$
Probably, you can expand the left-hand side using binomial expansion, but it would be tedious. How would you go around about it?
This is what I meant in the comment.
$${(1+a)^{25}}=\frac{3000a-300}{2500a-300} =\frac{30a-3}{25a-3}$$ Set $b=a+1$ $$b^{25}= \frac{30b - 33}{25b - 28}$$ $$b^{25} \left(25b - 28\right)=30b - 33 $$ $$25b^{26} - 28 b^{25} - 30b + 33 = 0 $$ Okay, it's not very nice, but at least the tedious binomial expansion was avoided. A numerical solver might be applied to estimate a root.