Geometric Sequences Section 3.2.1

Vocabulary Geometric Sequence: A sequence in which the ratio of any term to the previous term is constant. Common Ratio: The constant ratio between consecutive terms of a geometric sequence, denoted by r.

Investigation 1: Recall: An arithmetic sequence is a sequence in which the difference between two consecutive terms is constant. The constant difference between terms of an arithmetic sequence is denoted d and the explicit formula to find the nth term of a sequence is: an = a1 + d(n – 1).

Identify the next three terms of the arithmetic sequence, then write the explicit formula for the sequence: 3, 7, 11, 15, an = 3 + 4(n – 1) or an = 4n – 1 19, 23, 27, . . . Use the formula from example #1 to find the 27th term of the sequence. a27 = 3 + 4(27 – 1) = 107

In an arithmetic sequence, the terms are found by adding a constant amount to the preceding term. In a geometric sequence, the terms are found by multiplying each term after the first by a constant amount. This constant multiplier is called the common ratio and is denoted r. For each geometric sequence, identify the common ratio, r. 3. 2, 6, 18, 54, 162, . . . 4. 5, 50, 500, 5000, . . . 5. 3, , , , . . . -4, 24, -144, 864, -5184, . . . r = 3 r = 10 r = ½ r = -6

Tell whether the sequences is arithmetic, geometric or neither Tell whether the sequences is arithmetic, geometric or neither. For arithmetic sequences, give the common difference. For geometric sequences, give the common ratio.   7. 5, 10, 15, 20, 25, …. 8. 1, 1, 2, 3, 5, 8, 13, 21, … 9. 1, -4, 16, -64, 256, … 10. 512, 256, 128, 64, 32, … arithmetic; d = 5 neither geometric; r = -4 geometric; r = ½

Check for Understanding:   11. Find the first four terms of a geometric sequence in which a1 = 5 and r = -3. _____ , _____ , _____ , _____. 12. Find the missing term in the geometric sequence: -7, _______ , -28, 56, _______ , . . . 5 -15 45 -135 × -3 × -3 × -3 14 -112 × -2 × -2 56 ÷ -28 = -2 So, r = -2

Investigation 2: The explicit formula used to find the nth term of a geometric sequence with the first term a1 and the common ratio r is given by: an = a1∙ rn-1 Write a rule for the nth term of the sequence given. Then find a10.  

13. 1, 6, 36, 216, 1296, … Rule: an = 1∙6n-1 a10 = 1∙610-1 = 10077696    14. 14, 28, 56, 112, … Rule: an = 14∙2n-1 a10 = 14∙210-1 = 7168

Check for Understanding:   15. If a5 = 324 and r = -3, write the explicit formula for the geometric sequence and find a10. _____ , _____ , _____ , _____, 324 Rule: an = 4∙(-3)n-1 a10 = 4∙(-3)10-1 = -78732 4 -12 36 -108 OR ÷ -3 ÷ -3 ÷ -3 ÷ -3

16. If a3 = 18 and r = 3 write the explicit formula for the geometric sequence and find a10. Rule: an = 2∙(3)n-1 a10 = 2∙(3)10-1 = 39366

20. If r = 2 and a1 = 1 for a geometric sequence, Write a rule for the nth term of the sequence. b. Graph the first five terms of the sequence. (1, 1), (2, 2), (3, 4), (4, 8), (5, 16) What kind of graph does this represent? exponential