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To solve for the value of \( c \) in the given probability mass function (pmf) of the discrete random variable \( X \), we need to use the fact that the sum of all probabilities in a pmf must equal 1.

### Problem Recap:
The pmf is given as:
\[
p_X(2^{-k}) = P(X = 2^{-k}) = \frac{c}{k(k+1)}, \quad k = 2, 3, 4, \ldots
\]
where \( c \) is a positive constant.

### Step 1: Write the total probability condition
The total probability must sum to 1:
\[
\sum_{k=2}^{\infty} p_X(2^{-k}) = 1
\]

Substituting the pmf:
\[
\sum_{k=2}^{\infty} \frac{c}{k(k+1)} = 1
\]

### Step 2: Simplify the summation expression
Notice that:
\[
\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}
\]
This is a telescoping series.

So, the sum becomes:
\[
\sum_{k=2}^{\infty} \left(\frac{1}{k} - \frac{1}{k+1}\right)
\]

The telescoping nature means that most terms will cancel out, leaving:
\[
\left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \left(\frac{1}{4} - \frac{1}{5}\right) + \cdots
\]
This simplifies to:
\[
\frac{1}{2}
\]

### Step 3: Solve for \( c \)
Now, equate the sum to 1:
\[
c \times \frac{1}{2} = 1
\]
\[
c = 2
\]

### Final Answer:
The value of \( c \) is \( \boxed{2} \).

---

Would you like further details on any step? Here are five related questions to deepen your understanding:

1. How does the telescoping series help in simplifying summations?
2. What would the pmf look like for different values of \( k \)?
3. How is the constant \( c \) determined in other types of discrete distributions?
4. Can you derive the pmf of \( X \) if it had a different form for \( k \)?
5. How would the problem change if the summation range started from a different \( k \)?

**Tip:** Always remember that in probability, the total sum of the pmf must equal 1. This is a key concept in solving for unknown constants.

Let X be a discrete random variable with pmf p_X(2^{-k}) = P(X = 2^{-k}) = \frac{c}{k(k+1)}, k = 2, 3, 4, ..., where c is a positive constant. (a) Find the value of c.

Explore how to determine the constant c in the probability mass function of a discrete random variable. This problem involves understanding probability concepts, telescoping series, and the application of the total probability theorem. Suitable for undergraduate students.

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