Given that $$\sum_{r=1}^k \sin^{-1}\beta_r=0$$ for any $k\ge1$
A number $p$ is now defined such that $$p=\sum_{r=1}^k(\beta_r)^r$$
Then what is the value of the limit below? $$\lim_{x\rightarrow p} \frac{(1+x^2)^{\frac{1}{3}}-(1-2x)^{\frac{1}{4}}}{x+x^2}$$
This limit can be easily seen to be $\frac{1}{2}$ by putting $k=1$. Then we get $\beta=0$ which in turn implies $p=0$. Now the value of the limit becomes obvious.
But I am absolutely helpless in generalising and proving this result for arbitrary values of $k$.
Given that this given equation is true $\forall k \ge 1$, let it be true for $k=n$,
Now, it must be also true for $k=n+1$, you can easily subtract the two equations to get $\beta_n=0$.
Thus, you get $$\beta_r=0 ~\forall~ r \in \mathbb N$$
Implying $p=0$