Question

anne arundel county public schools | division of academics
7.4 geometric sequences
a ________________________ is a sequence where the ratio between consecutive terms is constant.
the ratio between the consecutive terms of a geometric sequence is called the ________________________.
the common ratio is found by dividing any term in a sequence by the ________________________ term. for example, the common ratio can be found:
$r = \frac{term\\ 2}{term\\ 1}$ or $\frac{term\\ 3}{term\\ 2}$ or $\frac{term\\ 6}{term\\ 5}$ or $\frac{term\\ 10}{term\\ 9}$ and so on.

recursive formula
a rule that uses a preceding term to find the next term only.
$g_n = a_{(n-1)} \cdot r$
for example,
multiply the 4th term by $r$ to find the 5th term.
multiply the 7th term by $r$ to find the 8th term.
multiply the 10th term by $r$ to find the 11th term.
and so on.

explicit formula
a rule to find any term in the sequence.
$g_n = a_1 \cdot r^{n-1}$
the explicit formula does not require you to know the preceding term.
you only need to know the first term and the common ratio, $r$.

1. is the sequence $-2, -4, -8, -16, ...$ geometric? if so, what is the common ratio?
2. is the sequence $3, 6, 9, 12, ...$ geometric? if so, what is the common ratio?

algebra 1 unit 7.4

Explanation:

Step1: Fill definition blanks

1. A geometric sequence is a sequence where the ratio between consecutive terms is constant.
2. The ratio between the consecutive terms of a geometric sequence is called the common ratio.
3. The common ratio is found by dividing any term in a sequence by the previous term.

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Step2: Analyze sequence 1 (-2,-4,-8,-16,...)

Calculate consecutive term ratios:
$\frac{-4}{-2}=2$, $\frac{-8}{-4}=2$, $\frac{-16}{-8}=2$
All ratios are equal, so it is geometric.

Step3: Analyze sequence 2 (3,6,9,12,...)

Calculate consecutive term ratios:
$\frac{6}{3}=2$, $\frac{9}{6}=1.5$, $\frac{12}{9}=\frac{4}{3}$
Ratios are not equal, so it is not geometric.

Answer:

1. Filled blanks:

• geometric sequence
• common ratio
• previous

1. For sequence $-2,-4,-8,-16,...$:

Yes, it is geometric. The common ratio is $2$.

1. For sequence $3,6,9,12,...$:

No, it is not geometric. There is no constant common ratio.