algebra 2 geometric sequences and series powerpoint 2012
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Find terms of a geometric sequence, including geometric means.
Find the sums of geometric series.
Objectives
Serena Williams was the winner out of 128 players who began the 2003 Wimbledon Ladies’ Singles Championship. After each match, the winner continues to the next round and the loser is eliminated from the tournament. This means that after each round only half of the players remain.
The number of players remaining after each round can be modeled by a geometric sequence. In a geometric sequence, the ratio of successiveterms is a constant called the common ratio r (r ≠ 1) . For the players remaining, r is ..
Recall that exponential functions have a commonratio. When you graph the ordered pairs (n, an) of ageometric sequence, the points lie on an exponentialcurve as shown. Thus, you can think of a geometricsequence as an exponential function with sequentialnatural numbers as the domain.
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Example 1A: Identifying Geometric Sequences
100, 93, 86, 79, ...
100, 93, 86, 79
Differences –7 –7 –7
Ratios 93 86 79 100 93 86
It could be arithmetic, with d = –7.
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Example 1B: Identifying Geometric Sequences
180, 90, 60, 15, ...
180, 90, 60, 15
Differences –90 –30 –45
It is neither.
3Ratios 1 1 1
2 4
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Example 1C: Identifying Geometric Sequences
5, 1, 0.2, 0.04, ...
5, 1, 0.2, 0.04
Differences –4 –0.8 –0.16
5Ratios 1 1 1
5 5
It could be geometric, with
Try 1a
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Differences
It could be geometric with
Ratios
1.7, 1.3, 0.9, 0.5, . . .
Try 1b
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
1.7 1.3 0.9 0.5
Differences –0.4 –0.4 –0.4
It could be arithmetic, with r = –0.4.
Ratio
Each term in a geometric sequence is the product of the previous term and the common ratio, giving the recursive rule for a geometric sequence.
an = an–1r nth term
Common ratio
First term
You can also use an explicit rule to find the nth term of a geometric sequence. Each term is the product of the first term and a power of the common ratio as shown in the table.
This pattern can be generalized into a rule for all geometric sequences.
Find the 7th term of the geometric sequence 3, 12, 48, 192, ....
Example 2: Finding the nth Term Given a Geometric Sequence
Step 1 Find the common ratio.
r = a2
a1
123
= 4=
Example 2 Continued
Step 2 Write a rule, and evaluate for n = 7.
an = a1 r n–1
a7 = 3(4)7–1
= 3(4096) = 12,288
The 7th term is 12,288.
General rule
Substitute 3 for a1,7 for n, and 4 for r.
Try 2a Continued
Step 2 Write a rule, and evaluate for n = 9.
an = a1 r n–1 General rule
The 9th term is .
Substitute for a1, 9 for
n, and for r.
0.001, 0.01, 0.1, 1, 10, . . .
Try 2b
Find the 9th term of the geometric sequence.
Step 1 Find the common ratio.
Try 2b Continued
Step 2 Write a rule, and evaluate for n = 9.
an = a1 r n–1
a9 = 0.001(10)9–1
= 0.001(100,000,000) = 100,000
The 7th term is 100,000.
General rule
Substitute 0.001 for a1,
9 for n, and 10 for r.
Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324.
Example 3: Finding the nth Term Given Two Terms
Step 1 Find the common ratio.
a5 = a3 r(5 – 3)
a5 = a3 r2
324 = 36r2
9 = r2
3 = r
Use the given terms.
Simplify.
Substitute 324 for a5 and 36 for a3.
Divide both sides by 36.
Take the square root of both sides.
Example 3 Continued
Step 2 Find a1.
Consider both the positive and negative values for r.
an = a1r n - 1
36 = a1(3)3 - 1
4 = a1
an = a1r n - 1
36 = a1(–3)3 - 1
4 = a1
General rule
Use a3 = 36 and r = 3.
or
Example 3 Continued
Step 3 Write the rule and evaluate for a8.
Consider both the positive and negative values for r.
an = a1r n - 1 an = a1r n - 1
Substitute a1 and r.
The 8th term is 8748 or –8747.
an = 4(3)n - 1
a8 = 4(3)8 - 1
a8 = 8748
an = 4(–3)n - 1
a8 = 4(–3)8 - 1
a8 = –8748
Evaluate for n = 8.
General rule
or
When given two terms of a sequence, be sure to consider positive and negative values for r when necessary.
Caution!
Try 3a
Find the 7th term of the geometric sequence with the given terms.
a4 = –8 and a5 = –40Step 1 Find the common ratio.
a5 = a4 r(5 – 4)
a5 = a4 r
–40 = –8r
5 = r
Use the given terms.
Simplify.
Substitute –40 for a5 and –8 for a4.
Divide both sides by –8.
Try 3a Continued
Step 2 Find a1.
an = a1r n - 1
–8 = a1(5)4 - 1
–0.064 = a1
General rule
Use a5 = –8 and r = 5.
Try 3a Continued
Step 3 Write the rule and evaluate for a7.
an = a1r n - 1
Substitute for a1 and r.
The 7th term is –1,000.
an = –0.064(5)n - 1
a7 = –0.064(5)7 - 1
a7 = –1,000
Evaluate for n = 7.
The indicated sum of the terms of a geometric sequence is called a geometric series. You can derive a formula for the partial sum of a geometric series by subtracting the product of Sn and r from Sn as shown.
Find the indicated sum for the geometric series.
Example 5A: Finding the Sum of a Geometric Series
S8 for 1 + 2 + 4 + 8 + 16 + ...
Step 1 Find the common ratio.
Example 5B: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
Step 1 Find the first term.
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