Holt McDougal Algebra 2 Geometric Sequences and Series Find terms of a geometric sequence, including geometric means. Find the sums of geometric series

Holt McDougal Algebra 2

Geometric Sequences and Series

Find terms of a geometric sequence, including geometric means.

Find the sums of geometric series.

Objectives



geometric sequencegeometric meangeometric series

Vocabulary



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, ...




Example 1B: Identifying Geometric Sequences

180, 90, 60, 15, ...




Example 1C: Identifying Geometric Sequences

5, 1, 0.2, 0.04, ...



Check It Out! Example 1a




1.7, 1.3, 0.9, 0.5, . . .

Check It Out! Example 1b




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 termCommon ratio

First term



0.001, 0.01, 0.1, 1, 10, . . .


Find the 9th term of the geometric sequence.



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) Use the given terms.



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

an = a1r n - 1

36 = a1(–3)3 - 1or



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

an = 4(3)n - 1 an = 4(–3)n - 1or



When given two terms of a sequence, be sure to consider positive and negativevalues for r when necessary.

Caution!




Find the 7th term of the geometric sequence with the given terms.

a4 = –8 and a5 = –40



a2 = 768 and a4 = 48


Find the 7th term of the geometric sequence with the given terms.



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 + ...




Example 5A: Finding the Sum of a Geometric Series

S8 for 1 + 2 + 4 + 8 + 16 + ...




Example 5A Continued

Step 2 Find S8 with a1 = 1, r = 2, and n = 8.

Sum

formula

Substitute.

Check Use a graphing calculator.



Example 5A Continued

Step 2 Find S8 with a1 = 1, r = 2, and n = 8.

Sum

formula

Substitute.




Example 5B: Finding the Sum of a Geometric Series




Example 5B: Finding the Sum of a Geometric Series


Step 1 Find the first term.



Example 5B Continued

Step 2 Find S6.Check Use a graphing calculator.

Sum

formula

Substitute.



Example 5B Continued

Step 2 Find S6.

= 1(1.96875) ≈ 1.97


Sum

formula

Substitute.




Find the indicated sum for each geometric series.

S6 for






S6 for



Check It Out! Example 5a Continued

Step 2 Find S6 with a1 = 2, r = , and n = 6.



Check It Out! Example 5a Continued

Step 2 Find S6 with a1 = 2, r = , and n = 6.

Substitute.

Sum formula









Step 1 Find the first term.



Step 2 Find S6.

Check It Out! Example 5b Continued



Step 2 Find S6.

Check It Out! Example 5b Continued



Lesson Quiz: Part I

1. Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.

2. Find the 8th term of the geometric sequence 1, –2, 4, –8, ….

3. Find the 9th term of the geometric sequence with a2 = 0.3 and a6 = 0.00003.



Lesson Quiz: Part II

4. Find the geometric mean of and 18.

5. Find the indicated sum for the geometric series

6. A math tournament begins with 81 students. Students compete in groups of 3, with 1 person from each trio going on to the next round until there is 1 winner. How many matches must be played in order to complete the tournament?

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Holt McDougal Algebra 2 Geometric Sequences and Series Find terms of a geometric sequence, including geometric means. Find the sums of geometric series