A sequence is an ordered
list of numbers.
The sum of the terms of a sequence is
called a series.
While some sequences are simply random values,
other sequences have a definite pattern that is
used to arrive at the sequence's terms.
Two such sequences are the arithmetic and geometric
sequences. Let's investigate the geometric sequence.
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Geometric Sequences |
MULTIPLY |
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If a sequence of values follows a pattern of
multiplying
a fixed amount (not
zero) times
each term to
arrive at the following term, it is referred to as a
geometric sequence.
The number multiplied each time is constant (always the same).
The fixed
amount multiplied is called the
common ratio, r,
referring to the fact that the ratio (fraction) of the
second term to the first term yields this common multiple.
To find the common ratio, divide the second term by the first
term.
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Notice the non-linear nature of the
scatter plot of the terms of a geometric sequence. The
domain consists of the counting numbers 1, 2, 3, 4, ... and the range
consists of the terms of the sequence. While
the x value increases by a constant value of one, the y
value increases by multiples of two (for this graph).
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Examples:
| Geometric
Sequence |
Common Ratio, r |
|
5, 10, 20, 40, ... |
r
= 2 |
multiply each term
by 2 to arrive
at the next term
or...divide a2
by a1 to find the common ratio, 2. |
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-11, 22, -44, 88, ... |
r = -2 |
multiply each term by -2 to arrive
at the next term
.or...divide a2
by a1 to find the common ratio, -2. |
 |
 |
multiply each term by 2/3 to arrive
at the next term or...divide
a2 by a1 to find the common
ratio, 2/3. |
Formulas used with
geometric sequences and
geometric series:
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To
find any term
of a geometric sequence:
where a1 is the first term of the
sequence,
r is the common ratio, n is the number
of the term to find. |
Note: a1
is often simply referred to as a.
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To find the
sum of a certain number of
terms of a geometric sequence:
where Sn is the sum of n
terms (nth partial sum),
a1 is the first term,
r is the common ration. |
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Examples:
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Question |
Answer |
1.
Find the common ratio for the sequence
 |
1. The
common ratio, r, can be found by dividing the second term
by the first term, which in this problem yields -1/2.
Checking shows that multiplying each entry by -1/2 yields the
next entry. |
2.
Find the common ratio for the sequence given by the formula
 |
2. The
formula indicates that 3 is the common ratio by its position in
the formula. A listing of the terms will also show what is
happening in the sequence (start with n = 1).
5, 15, 45, 135, ...
The list also shows the common ratio to be 3. |
3.
Find the 7th term of the sequence
2, 6, 18, 54, ... |
3. n = 7; a1
= 2, r = 3 
The seventh term is 1458.
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4.
Find
the 11th term of the sequence
 |
4. n = 11;
a1 = 1, r = -1/2  |
5.
Find a8
for the sequence
0.5, 3.5, 24.5, 171.5, ... |
5. n = 8; a1
= 0.5, r = 7
 |
6.
Evaluate using a
formula:
 |
6. Examine the summation
This is a geometric series with a common ratio of 3.
n = 5; a1 = 3, r = 3
 |
7. Find the sum of the
first 8 terms of the
sequence
-5, 15, -45, 135, ...
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7. The word
"sum" indicates a need for the sum formula.
n = 8; a1
= -5, r= -3
 |
| 8.
The third term of a
geometric sequence is 3 and the sixth term is 1/9. Find
the first term. |
8. Think of the sequence as "starting
with" 3, until you find the common ratio.
For this modified sequence: a1 = 3, a4
= 1/9,
n = 4
Now, work backward multiplying by 3 (or dividing by 1/3) to find the actual first term.
a1 = 27
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9.
A ball is dropped
from a height of 8 feet. The ball bounces to 80% of its
previous height with each bounce. How high (to the nearest
tenth of a foot) does the ball bounce on the fifth bounce?
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9. Set up a model drawing for each
"bounce".
6.4, 5.12, ___, ___, ___
The common ratio is 0.8.
Answer: 2.6 feet |
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Check out how to use your
TI-83+/84+ graphing calculator with
sequences and series.
Click here. |
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