I have to find the likelihood function for $X\sim \text{Bin}(n,0.5)$ where we know that $x=14$ and so $n \ge 14$. Several trials were conducted counting successes (14 of which there were) but the number of trials was forgotten, so we need to estimate $n$.
I know that the pmf of a Binomial random variable is $$P(X=x)=\binom{n}{x}\cdot p^x \cdot(1-p)^{n-x}$$ and in this case we have $$P(X=14)=\binom{n}{14}\cdot 0.5^{14} \cdot(1-0.5)^{n-14}=\binom{n}{14}\cdot0.5^{n}\,,$$ however I'm not sure this would be useful since for $n \ge 14$, the function seems to increase at an exponential rate that doesn't appear to have a maximum.
So I'm not sure using the pmf is appropriate - does anyone have any other ideas? Any help would be greatly appreciated.
No, if you read well your problem you have only to find a way to estimate the $n$ trials with 14 known successes among them
First Idea
using the hint of the text, I would approximate
$$B\Big(n;\frac{1}{2}\Big)\approx N\Big(\frac{n}{2}; \frac{n}{4} \Big)$$
Thus I would maximize
$$f(n)\propto n^{-\frac{1}{2}}e^{-\frac{1}{2n}(2x-n)^2 }$$
Finding
$$\hat{n}=\Bigg\lceil\frac{-1+\sqrt{1+16 x^2}}{2}\Bigg\rceil_{x=14}=28$$
Second idea, Method of Moments
Using a Negative Binomial, in average, how many trials are neeeded to get 14 successes?
$$\hat{n}=\mathbb{E}[N]=\frac{14}{\frac{1}{2}}=28$$
As you can see, the result is 28 in both cases