Question

1. given the sequence 2, 4, 8, 16, 32, ..., identify the following: arithmetic or geometric? common difference/ratio: explicit formula: recursive formula:

Explanation:

Step1: Classify the sequence

Check if terms have a common ratio:
$\frac{4}{2}=2$, $\frac{8}{4}=2$, $\frac{16}{8}=2$, $\frac{32}{16}=2$.
Since there is a common ratio (not a common difference), it is geometric.

Step2: Identify common ratio

From Step1, the common ratio $r=2$.

Step3: Derive explicit formula

Use geometric sequence explicit form $a_n = a_1 r^{n-1}$, where $a_1=2$, $r=2$:
$a_n = 2 \times 2^{n-1} = 2^n$

Step4: Derive recursive formula

Recursive form for geometric sequences: $a_n = r \times a_{n-1}$ with initial term $a_1=2$:
$a_1=2$, $a_n=2a_{n-1}$ for $n\geq2$

Answer:

Arithmetic or Geometric? Geometric
Common difference/ratio: $2$
Explicit Formula: $a_n = 2^n$
Recursive Formula: $a_1=2$, $a_n=2a_{n-1}$ for $n\geq2$