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From the image provided, it appears to be a question on calculating the sum of the first four terms of a geometric sequence. Let’s go through the details given and the solution steps based on the formula used in the image.

### Given Information
- **First term, \( a = 300 \)**
- **Common ratio, \( r = \frac{450}{300} = \frac{3}{2} \)**
- **Number of terms, \( n = 4 \)**

### Formula for the Sum of a Geometric Sequence
The sum of the first \( n \) terms of a geometric sequence is given by:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]

### Calculating \( S_4 \)
To find \( S_4 \), substitute \( a = 300 \), \( r = \frac{3}{2} \), and \( n = 4 \) into the formula:

\[
S_4 = 300 \cdot \frac{\left(\frac{3}{2}\right)^4 - 1}{\frac{3}{2} - 1}
\]

#### Step 1: Calculate \( \left(\frac{3}{2}\right)^4 \)
\[
\left(\frac{3}{2}\right)^4 = \frac{81}{16}
\]

#### Step 2: Substitute back
\[
S_4 = 300 \cdot \frac{\frac{81}{16} - 1}{\frac{3}{2} - 1}
\]

#### Step 3: Simplify the terms in the numerator
\[
\frac{81}{16} - 1 = \frac{81 - 16}{16} = \frac{65}{16}
\]

#### Step 4: Simplify the denominator
\[
\frac{3}{2} - 1 = \frac{1}{2}
\]

#### Step 5: Substitute into the expression
\[
S_4 = 300 \cdot \frac{\frac{65}{16}}{\frac{1}{2}}
\]

#### Step 6: Divide by \( \frac{1}{2} \) (which is equivalent to multiplying by 2)
\[
S_4 = 300 \cdot \frac{65}{16} \cdot 2 = 300 \cdot \frac{130}{16} = 300 \cdot \frac{65}{8}
\]

#### Step 7: Calculate the final answer
\[
S_4 = 300 \cdot 8.125 = 2437.5
\]

### Final Answer
The sum of the first four terms, \( S_4 \), is \( 2437.5 \).

Would you like further details on any specific step? 

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Here are some related questions to expand on this topic:
1. How do we determine if a sequence is geometric or arithmetic?
2. What changes in the formula if the common ratio \( r \) is less than 1?
3. How do we find the sum of an infinite geometric series?
4. What happens to the sum formula when \( r = 1 \)?
5. How can geometric sequences be applied in real-life scenarios?

**Tip:** In a geometric series, if the common ratio \( r \) is greater than 1, the terms grow rapidly, leading to larger sums for higher values of \( n \).

Calculate the sum of the first four terms of the geometric sequence 300 + 450 + 675 + ...

Learn how to find the sum of the first four terms of a geometric sequence starting with 300, 450, 675 using the sum formula for geometric series. This problem is suitable for students in Grades 10-12 and covers important concepts such as common ratio and the sum of a geometric sequence.

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