Lesson 10-7 Geometric Sequences

Key Concept A geometric sequence is a sequence in which each term after the nonzero term is found by multiplying the previous term by a constant called the common ratio, where r  0, 1. Symbols: a, ar, ar2, ar3, …(a  0; r  0, 1) Examples: 1, 3, 9, 27, 81,… Recognize Geometric Sequences Determine if the sequence is geometric. 0,5, 10, 15, 20,… Add 5 to previous, this is arithmetic, not geometric. 1,5,25, 125, 625,… Multiply by five. This is geometric. Determine whether each sequence is geometric. 1, 4, 16, 64, 256, … 1, 3, 5, 7, 9, 11, …

Continue Geometric Sequence Find the next three terms in each geometric sequence. 4, -8, 16, … Divide the second term by the first. The common factor is -2. Use the information to find the next three terms. -32, 64, -128 60, 72, 86.4, … Divide the second term by the first. The common factor is 1.2. Use the information to find the next three terms. 103.68, 124.416, 149.2992 Find the next three terms in each sequence. 20, -28, 39.2, … 64, 48, 36… -54.88, 76.832, -107.5648 27, 20.25, 15.1875

Use Geometric Sequence to Solve a Problem Year Population 2000 15,500,000 2001 15,500,000(1.03) or 15,965,000 2002 15,965,000(1.03) or 16,4443,950 2003 16,4443,950(1.03) or 16,937,269 Madagascar’s population has been increasing at an average annual rate of 3%. In 2000, the population was 15,500,000. Determine Madagascar’s population in 2001, 2002, and 2003. The population is a geometric sequence in which the first term is 15,500,000 and the common ratio is 1.03. The population of the African country of Liberia was about 2,900,000 in 1999. If the population grows at a rate of 5% per year, what will the population be in the years 2003, 2004, and 2005? 2003 - 3,524,968; 2004 - 3,701,217; 2005 - 3,866,277

Formula for the Nth term of a Geometric Sequence. The nth term an of a geometric sequence with the term a1 and a common ratio of r is given by an  a1  r n-1 Find the sixth term of a geometric sequence in which a1 = 3 and r = -5. an  a1  r n-1 Formula for the nth term of a geometric sequence. a6  3  (-5) 6-1 n = 6, a1 = 3, and r = -5 a6  3  (-5)5 6-1 = 5 a6  3  (-3125) (-5)5 = -3125 a6 = -9375 3  (-3125) = -9375 The sixth term of the geometric sequence is -9375. Find the eigth term of a geometric sequence in which a1 = 7 and r = 3. 15,309

Find the geometric mean in the sequence 2, ___, 18. Missing terms between two nonconsecutive terms in a geometric sequence are called geometric means. Find Geometric Means Find the geometric mean in the sequence 2, ___, 18. In the sequence, a1 = 2 and a3 = 18. To find a2, you must first find r. an = a1  r n-1 Formula for the nth term of a geometric sequence. a3 = a1  r 3-1 n = 3 18 = 2  r 2 a3 = 18, and a1 = 2 Divide each side by 2 9 = r2 Simplify 3 = r Take the square root of each side. If r = 3, the geometric mean is 2(3) or 6. If r = -3, the geometric mean is 2(-3) or -6. Therefore, the geometric mean is 6 or -6. Find the geometric mean in the sequence 7, ___, 112. 28