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This problem involves a geometric series where we are given the sum of the first three terms and the sum of the first four terms. Let's denote the following:

- Let the first term of the geometric series be \( a \),
- The common ratio be \( r \).

### Step 1: Sum of the first three terms
The sum of the first three terms of a geometric series can be written as:

\[
S_3 = a + ar + ar^2
\]

From the problem, we know:

\[
a + ar + ar^2 = 14.25
\]

Factoring out \( a \):

\[
a(1 + r + r^2) = 14.25 \quad \text{(Equation 1)}
\]

### Step 2: Sum of the first four terms
The sum of the first four terms can be written as:

\[
S_4 = a + ar + ar^2 + ar^3
\]

From the problem, we know:

\[
a + ar + ar^2 + ar^3 = 24.375
\]

Factoring out \( a \):

\[
a(1 + r + r^2 + r^3) = 24.375 \quad \text{(Equation 2)}
\]

### Step 3: Solving the system of equations
We now have the following two equations:

1. \( a(1 + r + r^2) = 14.25 \)
2. \( a(1 + r + r^2 + r^3) = 24.375 \)

We can divide Equation 2 by Equation 1 to eliminate \( a \) and find \( r \). Let's compute this.

Let me proceed with the calculation.It seems I can't directly divide the equations like that. Instead, let's manipulate the equations algebraically to eliminate \( a \). I'll divide the expressions on both sides of the equations manually and then solve for the common ratio \( r \).

We have:

\[
\frac{a(1 + r + r^2 + r^3)}{a(1 + r + r^2)} = \frac{24.375}{14.25}
\]

This simplifies to:

\[
\frac{1 + r^3}{1} = \frac{24.375}{14.25}
\]

Let me now compute \( r \).The real solution for the common ratio \( r \) is approximately:

\[
r \approx 0.892
\]

Now that we have \( r \), we can substitute this back into one of the original equations (say, Equation 1) to solve for \( a \).

Let me calculate the value of \( a \).The first term of the geometric series \( a \) is approximately:

\[
a \approx 5.30
\]

### Final answers:
- The first term \( a \) is approximately 5.30.
- The common ratio \( r \) is approximately 0.892.

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How do you calculate the sum of the first \( n \) terms of a geometric series?
2. What happens to a geometric series when the common ratio is greater than 1?
3. Can you find the 5th term of this geometric series using the values of \( a \) and \( r \)?
4. What is the formula for the sum of an infinite geometric series?
5. How does the sum of a geometric series change if the common ratio is negative?

**Tip:** The sum of a finite geometric series is a powerful tool to compute large series quickly, especially in sequences where multiplication grows or shrinks exponentially.

In a geometric series, the sum of the first three terms is 14.25 and the sum of the first four terms is 24.375. Find the first term and the common ratio.

Learn how to find the first term and common ratio of a geometric series where the sum of the first three terms is 14.25 and the sum of the first four terms is 24.375. This solution includes key algebraic and geometric series formulas, suitable for students in Grades 10-12.

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