The first, second and third terms of an arithmetic progression are a, 2a and a
2
respectively, where a
is a positive constant.
Find the sum of the first 50 terms of the progression.

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.

Explore how to calculate the sum of the first 50 terms of an arithmetic progression with step-by-step guidance. Understand the concept of arithmetic progressions, including the formula for the sum of terms, and solve related problems effectively. Suitable for Grades 10-12 students and beyond.

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