Let's start by analyzing the problem. We are given that the first three terms of an arithmetic progression (AP) are $a$, $2a$, and $a^2$, where $a$ is a positive constant.

Step 1: Finding the common difference

In an arithmetic progression, the difference between consecutive terms is constant. Therefore, the common difference $d$ can be found as follows:

$d = 2a - a = a$

Similarly, the second difference between the second and third terms should also equal the common difference:

$d = a^2 - 2a$

We now have two expressions for $d$:

$a = a^2 - 2a$

Step 2: Solving the equation for $a$

To solve for $a$, we set up the equation:

$a^2 - 2a - a = 0$

This simplifies to:

$a^2 - 3a = 0$

Factoring the equation:

$a(a - 3) = 0$

This gives us two possible solutions:

$a = 0 \quad \text{or} \quad a = 3$

Since $a$ is a positive constant, we discard $a = 0$ and take $a = 3$.

Step 3: Determining the sum of the first 50 terms

Now that we know $a = 3$, the first three terms of the arithmetic progression are:

$3, 6, 9, \ldots$

The common difference $d$ is $3$.

The sum of the first $n$ terms of an arithmetic progression is given by the formula:

$S_n = \frac{n}{2} \times [2a_1 + (n-1)d]$

For the first 50 terms:

$S_{50} = \frac{50}{2} \times [2(3) + (50-1)(3)]$

Simplifying this:

$S_{50} = 25 \times [6 + 147]$

$S_{50} = 25 \times 153 = 3825$

So, the sum of the first 50 terms of the progression is $3825$.

Would you like a more detailed explanation or have any other questions?

Related Questions:

1. What is the 50th term of the arithmetic progression?
2. How does the sum change if $a$ were a different value?
3. What is the formula for the sum of the first $n$ terms of an arithmetic progression?
4. How can we verify that our value of $a = 3$ is correct?
5. How does the common difference affect the sum of the first 50 terms?

Tip:

In arithmetic progressions, understanding how to derive the common difference is key to solving most related problems effectively.