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.