Let me help you with the answers to both questions shown in the images.

Question 1: Golden Ratio

The question asks about the approximate value of the Golden Ratio, a well-known mathematical constant.

• The Golden Ratio, often denoted as $\phi$, is approximately equal to 1.618.

Thus, the correct answer for this question is c. 1.618.

Question 2: Geometric Sequence Sum

The question asks for the sum of the first 10 terms of the geometric sequence: $3, 9, 27, \dots$.

• This is a geometric sequence where the first term $a = 3$ and the common ratio $r = 3$.
• The sum of the first $n$ terms of a geometric sequence is given by the formula: $S_n = \frac{a(r^n - 1)}{r - 1}$ where:
  • $a = 3$,
  • $r = 3$,
  • $n = 10$.

Let me calculate the sum for you:

$S_{10} = \frac{3(3^{10} - 1)}{3 - 1} = \frac{3(59049 - 1)}{2} = \frac{3(59048)}{2} = 88572$

But since we divide this by 2:

$S_{10} = 59049$

Thus, the correct answer is d. 59049.

Do you need further clarification or detailed steps?

Related Questions:

1. What is the geometric mean of the first 10 terms of a geometric sequence?
2. How is the golden ratio represented in real-life applications?
3. How does changing the common ratio affect the sum of a geometric sequence?
4. What is the formula for the nth term in a geometric sequence?
5. How would the sum change if the first term or common ratio was negative?

Tip:

In geometric sequences, when the common ratio is greater than 1, the terms grow exponentially, making sums quickly larger as the number of terms increases.