Question

r = \frac{1}{3} to find f(5), f(6), and f(7).

f(5) = f(4) \cdot \frac{1}{3}
f(5) = 36 \cdot \frac{1}{3}
f(5) = 12

f(6) = f(5) \cdot \frac{1}{3}
f(6) = 12 \cdot \frac{1}{3}
f(6) = 4

f(7) = f(6) \cdot \frac{1}{3}
f(7) = 4 \cdot \frac{1}{3}
f(7) = \frac{4}{3}

the common ratio of the geometric sequence is ___ and the next three terms of the sequence are _, _, and ___.

find the 6^{th} term of the geometric sequence represented below.
3, 1, 0.5, 0.25, ...
\frac{0.25}{0.5} = 0.5, \frac{0.5}{1} = 0.5; the common ratio is r = 0.5

f(4) = 0.25(0.5)
f(4) = 0.125

f(5) = 0.125(0.5)
f(5) = 0.0625

f(6) = 0.0625(0.5)
f(6) = 0.03125

the 6^{th} term is _____.

Explanation:

Identify the common ratio

The calculations show:

$$ r = \frac{1}{3} $$

$$ f(5) = f(4) \cdot \frac{1}{3} = 36 \cdot \frac{1}{3} = 12 $$

$$ f(6) = f(5) \cdot \frac{1}{3} = 12 \cdot \frac{1}{3} = 4 $$

$$ f(7) = f(6) \cdot \frac{1}{3} = 4 \cdot \frac{1}{3} = \frac{4}{3} $$

Thus, the common ratio is \(\frac{1}{3}\).

Determine the next three terms

The next three terms calculated are:

$$ f(5) = 12 $$

$$ f(6) = 4 $$

$$ f(7) = \frac{4}{3} $$

Answer:

The common ratio of the geometric sequence is <blank>\(\frac{1}{3}\)</blank> and the next three terms of the sequence are <blank>\(12\)</blank>, <blank>\(4\)</blank>, and <blank>\(\frac{4}{3}\)</blank>.