10.4 Geometric Sequences and Series KPIs 1 10.4 Geometric Sequences and Series STARTER Solve the equations:(No decimal approximation. Give exact answers.) 1 x5 = 2 2 x 5 = 243a 10 3 64x 7 = 1 2 4 5x = 2 2 10.4 Geometric Sequences and Series NOTES 10.4.1 Find the common ratio in a geometric sequence given either two consecutive terms, two nonconsecutive terms, or the equation of the nth term 1. Find the common ratio given two consecutive terms of a geometric sequence: Two consecutive terms of a sequence are represented as: The common ratio “r” can be found by using : Examples: a) Find the common ratio of the geometric sequence if 5 and 15 are two consecutive terms of the sequence. b) The third term of a geometric sequence is -30, and the fourth term is 5. What is the common ratio? c) If the eleventh term and the twelfth term of a geometric sequence are -3 and 27 respectively, what is the common ratio? d) In a geometric sequence, a19 = 72 and a20 = 36. What is the common ratio? 3 10.4 Geometric Sequences and Series NOTES 10.4.1 Find the common ratio in a geometric sequence given either two consecutive terms, two nonconsecutive terms, or the equation of the nth term 2. Find the common ratio given two non-consecutive terms of an geometric sequence: Two non-consecutive terms of a sequence are represented as: The common ratio can be found by using : Examples: a) The rst term of a geometric sequence is 7, and the fth term is 112. Determine the common ratio. b) The second term of a geometric sequence is common ratio? fi fi 4 1 , and the seventh term is 8. What is the 4 10.4 Geometric Sequences and Series NOTES 10.4.1 Find the common ratio in a geometric sequence given either two consecutive terms, two nonconsecutive terms, or the equation of the nth term 3. Find the common ratio given the equation of the nth term: The nth term of an geometric sequence is of the form: The common ratio can be found by using : Step 1: Find ______ Step 2: Find ______ Step 3: Then r = ______ a) In a geometric sequence, the equation of the nth term is given by an What is the common ratio? b) The nth term of a geometric sequence is given by an = 3(4)n−1. = 52n−1. Find the common ratio. 2n c) If the equation of the nth term of a geometric sequence is an = , what is the 3n−1 common ratio? 5 10.4 Geometric Sequences and Series NOTES 10.4.2 Write the equation of the nth term of an geometric sequence given a list of consecutive terms or a term and the common ratio 6 10.4 Geometric Sequences and Series NOTES 10.4.3 Find the value of a specific nth term of an geometric sequence given a list of consecutive terms or a term and the common ratio 7 10.4 Geometric Sequences and Series NOTES 10.4.4 List the geometric means of an geometric sequence between two given terms 8 10.4 Geometric Sequences and Series NOTES 10.4.5 Identify the geometric series Recall Series: Write 3 examples of geometric sequences and the corresponding series for each sequence in the table below. Also write the indicated Partial Sum. Sequence Series Partial Sum S3 = 1 2 S5 = 3 S1 9 10.4 Geometric Sequences and Series NOTES 10.4.6 Find the partial sum of a geometric series (Sn) given either the initial term and the nth term or the initial term and the number of terms 10 10.4 Geometric Sequences and Series NOTES 10.4.7 Apply sigma notation to find the sum of a geometric series Find the first term a1 = The common ratio, r = The number of terms in series, n = Sn = Find each sum. 11 10.4 Geometric Sequences and Series NOTES 10.4.8 Find missing terms in a geometric series given a combination of the sum, first term, nth term, the common ratio and the number of terms 12 10.4 Geometric Sequences and Series NOTES 10.4.9 Solve real world applications related to finite geometric series and their sums 13 10.3 Geometric Sequences and Series PRACTICE 14 10.3 Geometric Sequences and Series PRACTICE 15 10.3 Geometric Sequences and Series PRACTICE 16