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GEOMETRIC SEQUENCES<BR>
GEOMETRIC SEQUENCES
A geometric sequence is a number sequence in which each successive
term may be found by multiplying by the same number.
If the number by which each term is multiplied is greater than 1,
the sequence grows at an increasing rate.
The figure above illustrates a geometric sequence resulting from repeatedly multiplying by 3.
If each term is multiplied by a positive number that is less than 1,
the sequence shrinks at a decreasing rate.
The figure below illustrates a geometric sequence resulting from repeatedly multiplying by 1/2.
All ratios of successive terms in a geometric sequence are the same.
Called the common ratio of the sequence,
it is the number by which each term is multiplied to get the next.
The common ratio of the left-hand sequence is
3/1 = 9/3 = 27/9 = 3
and the common ratio of the right-hand sequence is
8/16 = 4/8 = 2/4 = 1/2
QUESTIONS
The following number sequences appear on Babylonian cuneiform tabets of about 1800 B.C.
1 3 9 27 81 ...
and
1 4 16 64 256 ...
- How can successive terms of the first sequence be found?
- How can successive terms of the second sequence be found?
The figure below shows three stacks of poker chips.
The stacks are the first three terms of a geometric sequence.
- What are the first three terms?
- What is their common ratio?
- What are the next three terms of the sequence?
When a piano is tuned, the first note to be tuned is the A above middle C.
It has a frequency of 440 cycles per second.
Then the other seven A’s on the keyboard are tuned so that their frequencies form a geometric sequence.
? ? ? ? 440 880 ? ?
- What is the common ratio of this sequence?
- Find the frequencies of the other A’s.
- Write all eight terms of the sequence and then cross out every second term.
Do the remaining terms form a geometric sequence?
When a rubber ball is dropped from a height of 1280 centimeters,
the heights in centimeters of the first few bounces form a geometric sequence:
960 720 ? ?
- What is the common ratio of this sequence?
- Find the heights of the third and fourth bounces.
United States currency notes have been printed in the twelve denominations shown below.
Some of the denominations form the geometric sequence
1 10 100 ...
- What is the common ratio of this sequence?
- How many terms of this sequence are denominations of currency?
Compare the following patterns of the terms of an arithmetic sequence and a geometric sequence.
An arithmetic sequence: 2 5 8 11 14 ...
t1 = 2 = 2 + 3 x 0 = 2
t2 = 2 + 3 = 2 + 3 x 1 = 5
t3 = 2 + 3 + 3 = 2 + 3 x 2 = 8
t4 = 2 + 3 + 3 + 3 = ? = 11
t5 = 2 + 3 + 3 + 3 + 3 = ? = 14
A geometric sequence: 2 6 18 54 162 ...
t1 = 2 = 2 x 30 = 2
t2 = 2 x 3 = 2 x 31 = 6
t3 = 2 x 3 x 3 = 2 x 32 = 18
t4 = 2 x 3 x 3 x 3 = ? = 54
t5 = 2 x 3 x 3 x 3 x 3 = ? = 162
- What expressions should be written in the indicated spaces for t4 and t5 to complete the pattern in the arithmetic sequence?
- What expressions should be written in the indicated spaces for t4 and t5 to complete the pattern in the geometric sequence?
- Use the pattern for the arithmetic sequence to write an expression for its 10th term.
- Find the value of that term.
- Use the pattern for the geometric sequence to write an expression for its 10th term.
- Use a calculator to find the value of that term.
Write expressions for the indicated terms of the following geometric sequences.
- The 11th term of 7 35 175 875 ...
- The 50th term of 2 20 200 2,000 ...
- The 123rd term of 3 24 192 1,536 ...
Suppose the first term of a geometric sequence is t1 and its common ratio r.
Expressions for its first three terms are
t1 = t1
t2 = t1 • r
t3 = t1 • r2
Notice the relationship between the index of the term and the ratio's exponent.
Write expressions for
- the 8th term, t8 = _______.
- the nth term, tn = _______.
- the n+1th term, tn+1 = _______.
- Multiply the equation for the nth term by r, and simplify it.
- When you simplify it, is it equivalent to the n+1th term, tn+1?
- If it is equivalent, did you prove the nth + 1 term is true for all integers? Why?
- If it is not equivalent, what does that prove about the formula for the nth term?
- What method of reasoning did you use to answer the previous two questions? Inductive or deductive?
- What method of reasoning did you use to find the nth term? Inductive or deductive?
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Last Modified 12/9/08 7:54 AM
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