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?

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 $n$.