Is it possible to extract a variable out of a bounded sum of binomial coefficients I've got a problem that looks like $$y = \sum\limits^x_{i=0} { a \choose i } b^i$$ where $a, b$ are constant. I'd like to rewrite this equation to define $x$ in terms of $y$, to directly compute the $x$ that belong to certain values of $y$. If I use the Binomial identity $(1 + b)^a = \sum^\infty_{i=0} { a \choose i}b^i$, I can write the problem as $$y = \sum\limits^x_{i=0} { a \choose i } b^i= \sum^\infty_{i=0} {a \choose i}b^i - \sum^\infty_{i=x+1} {a \choose i}b^i = (1+b)^a - \sum^\infty_{i=x+1} {a \choose i}b^i $$ but that doesn't get me much further...

Set $\dfrac p{1-p}=b$ and write

$$y=\sum_{i=0}^x\binom aib^i=\sum_{i=0}^x\binom ai\frac{p^i}{(1-p)^i}=\frac 1{(1-p)^a}\sum_{i=0}^x\binom ai p^i(1-p)^{a-i}.$$

Then $y$ is a scaled cumulative binomial distribution, which can be denoted $(1-p)^aF(x;n,p)$.

The inverse can be expressed as

$$x=F^{-1}\left(\frac y{(1-p)^a};n,p\right)$$ and there is no explicit formula.

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If what you are after is a computation algorithm (which I suspect), a possible way is to precompute and store the partial sums up to the maximum $x$ and answer queries by dichotomic search.