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Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

Consider the given geometric sequence:

[tex]\[ 16, 32, 64, 128, \ldots \][/tex]

Given that the sequence is represented by the function [tex]\( f \)[/tex], complete the statements.

The common ratio is [tex]\(\square\)[/tex]

[tex]\[ f(1) = \square \][/tex]

[tex]\[ f(5) = \square \][/tex]



Answer :

GinnyAnswer
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]

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