To solve the given problem, let's analyze the geometric sequence step-by-step.
We are given the sequence:
[tex]\[ 16, 32, 64, 128, \ldots \][/tex]
1. Common Ratio:
The common ratio [tex]\( r \)[/tex] of a geometric sequence is found by dividing any term in the sequence by the previous term. Using the first two terms:
[tex]\[
r = \frac{a_2}{a_1} = \frac{32}{16}
\][/tex]
Simplifying this fraction:
[tex]\[
r = 2.0
\][/tex]
So the common ratio is [tex]\( 2.0 \)[/tex].
2. Finding [tex]\( f(1) \)[/tex]:
The function [tex]\( f \)[/tex] represents our geometric sequence. The first term in the sequence is given as [tex]\( a_1 = 16 \)[/tex].
Therefore:
[tex]\[
f(1) = 16.0
\][/tex]
3. Finding [tex]\( f(5) \)[/tex]:
For the [tex]\( n \)[/tex]-th term in a geometric sequence, we use the formula:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
To find [tex]\( f(5) \)[/tex]:
[tex]\[
f(5) = 16 \cdot 2^{(5-1)}
\][/tex]
Simplify the exponent:
[tex]\[
f(5) = 16 \cdot 2^4
\][/tex]
Compute [tex]\( 2^4 \)[/tex]:
[tex]\[
2^4 = 16
\][/tex]
Now multiply:
[tex]\[
f(5) = 16 \cdot 16 = 256.0
\][/tex]
So, the completed statements are:
- The common ratio is [tex]\( 2.0 \)[/tex]
- [tex]\( f(1) = 16.0 \)[/tex]
- [tex]\( f(5) = 256.0 \)[/tex]
Hence, the final answers are:
- The common ratio is [tex]\( \boxed{2.0} \)[/tex]
- [tex]\( f(1) = \boxed{16.0} \)[/tex]
- [tex]\( f(5) = \boxed{256.0} \)[/tex]
