Question

1. which explicit formula is equivalent to \\(a_1 = 1\\), \\(a_n = 4a_{n-1}\\)?

\\(a_n = 1 + (n-1)4\\)

\\(a_n = 1(4)^{n-1}\\)

\\(a_n = 4(4)^{n-1}\\)

\\(a_n = 4(1)^{n-1}\\)

Explanation:

⚡ Using what you learned: Recursive and Explicit Formulas

Step 1: Identify the sequence type

Given recursive formula:

$$ a_1 = 1 $$

$$ a_n = 4a_{n-1} $$

Since each term is found by multiplying the previous term by a constant factor of \( 4 \), this is a geometric sequence with:

• First term \( a_1 = 1 \)
• Common ratio \( r = 4 \)

Step 2: Write the explicit formula

The general explicit formula for a geometric sequence is:

$$ a_n = a_1(r)^{n-1} $$

Substitute \( a_1 = 1 \) and \( r = 4 \):

$$ a_n = 1(4)^{n-1} $$

Answer:

\( a_n = 1(4)^{n-1} \)