Question

write a rule for the ( n )th term of the sequence.
(the image shows a graph with points (1, 0.5), (2, 1), (3, 2), (4, 4), (5, 8) plotted on a coordinate grid with ( n ) on the x - axis and ( a_n ) on the y - axis. we need to find ( a_n = ) )

Explanation:

Step1: Identify sequence type

Check ratio of consecutive terms:
$\frac{1}{0.5}=2$, $\frac{2}{1}=2$, $\frac{4}{2}=2$, $\frac{8}{4}=2$. Common ratio $r=2$, so it's geometric.

Step2: Recall geometric sequence formula

The general rule for a geometric sequence is $a_n = a_1 r^{n-1}$, where $a_1$ is the first term.

Step3: Substitute known values

Here, $a_1=0.5=\frac{1}{2}$ and $r=2$. Substitute into the formula:
$a_n = \frac{1}{2} \times 2^{n-1}$

Step4: Simplify the expression

Use exponent rules: $\frac{1}{2}=2^{-1}$, so $a_n = 2^{-1} \times 2^{n-1} = 2^{n-2}$

Answer:

$a_n=2^{n-2}$