How to determine Y(n)

Question

A random variable $x$ from the set $\{1, 2, ... ,n\}. $ Let $x$ has distribution function $f(k) = Y(n) · g^k$ where $g$ is a fixed number within $0$ and $1$. Find $Y(n)$ which is a constant term in terms of n.

I do not know how to determine $Y(n)$. Can I integrate $f(k)$ ? Thank You.

Answer 1

Motivated by $P(x\in \lbrace 1,\dots, n \rbrace)=1$, meaning that $1 = \sum_{k=1}^n P(x=k)$

You should sum over all possibles values of $k$ :

$$1 = \sum_{k=1}^n Y(n)g^k = Y(n) \sum_{k=1}^n g^k$$

Then, you have :

$$Y(n) = \frac{1}{\sum_{k=1}^n g^k}$$

Finding the exact value of the sum is not difficult, I let you finish.

Answer 2

Hint: the PMF must sum to $1$. (Instead of integrating $f(k)$, try summing it!)