Question

1. what are the first five terms in the recursive sequence defined by the following?

\\(a_1 = 150\\)
\\(a_n = 0.85a_{n-1}\\)

\\(\bigcirc\\) \\(\\{0.85, 150.85, 300.85, 450.85, 600.85\\}\\)
\\(\bigcirc\\) \\(\\{150, 22.5, 3.375, 0.50625, 0.0759375\\}\\)
\\(\bigcirc\\) \\(\\{150, 127.5, 108.375, 92.11875, 78.3009375\\}\\)
\\(\bigcirc\\) \\(\\{0.15, 22.5, 3.375, 506.250, 75,937,500\\}\\)

Explanation:

⚡ Using what you learned: Geometric Sequences

Step 1: Identify the first term

The first term is given directly in the definition:

$$ a_1 = 150 $$

Step 2: Calculate the second term

Multiply the first term by \( 0.85 \):

$$ a_2 = 0.85 \cdot a_1 = 0.85 \cdot 150 = 127.5 $$

Step 3: Calculate the third term

Multiply the second term by \( 0.85 \):

$$ a_3 = 0.85 \cdot a_2 = 0.85 \cdot 127.5 = 108.375 $$

Step 4: Calculate the fourth and fifth terms

Multiply each subsequent term by \( 0.85 \):

$$ a_4 = 0.85 \cdot a_3 = 0.85 \cdot 108.375 = 92.11875 $$

$$ a_5 = 0.85 \cdot a_4 = 0.85 \cdot 92.11875 = 78.3009375 $$

Answer:

\( \{150, 127.5, 108.375, 92.11875, 78.3009375\} \) (the third option)