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Over Lesson –Minute Check 2 Write an explicit formula and a recursive formula for finding the nth term of the arithmetic sequence 2, 7, 12, 17, …. A.a n = 5n – 3; a 1 = 2, a n = a n – B.a n = 5n; a 1 = 2, a n = a n – C.a n = 5n + 2; a 1 = 7, a n = a n – D.a n = 7n – 2; a 1 = 7, a n = a n – 1 + 5

Over Lesson –Minute Check 2 Write an explicit formula and a recursive formula for finding the nth term of the arithmetic sequence 2, 7, 12, 17, …. A.a n = 5n – 3; a 1 = 2, a n = a n – B.a n = 5n; a 1 = 2, a n = a n – C.a n = 5n + 2; a 1 = 7, a n = a n – D.a n = 7n – 2; a 1 = 7, a n = a n – 1 + 5

Then/Now You found terms and means of arithmetic sequences and sums of arithmetic series. (Lesson 10-2) Find nth terms and geometric means of geometric sequences. Find sums of n terms of geometric series and sums of infinite geometric series.

Vocabulary geometric sequence common ratio geometric means geometric series

Example 1 Geometric Sequences A. Determine the common ratio and find the next three terms of the geometric sequence –6, 9, –13.5, …. First, find the common ratio. a 2 ÷ a 1 = 9 ÷ (–6) or –1.5 Find the ratio between two pairs of consecutive terms to determine the common ratio. a 3 ÷ a 2 = –13.5 ÷ 9 or –1.5 The common ratio is –1.5. Multiply the third term by –1.5 to find the fourth term, and so on.

Example 1 Answer: –1.5; 20.25, –30.375, Geometric Sequences a 4 = (–13.5)(–1.5) or a 5 = 20.25(–1.5) or – a 6 = (–30.375)(–1.5) or The next three terms are 20.25, –30.375, and

Example 1 Geometric Sequences B. Determine the common ratio and find the next three terms of the geometric sequence 243n – 729, –81n + 243, 27n – 81, …. First, find the common ratio. a 2 = –81n a 1 = 243n – 729 Factor.Simplify.

Example 1 Geometric Sequences Simplify. Factor. a 3 = 27n – 81 a 2 = –81n The common ratio is. Multiply the third term by to find the fourth term, and so on.

Example 1 Geometric Sequences The next three terms are –9n + 27, 3n – 9, and –n + 3. Answer: ; –9n + 27, 3n – 9, –n + 3

Example 1 Determine the common ratio and find the next three terms of the geometric sequence 24, 84, 294, …. A.–3; –882, 2646, –7938 B.3.5; 297.5, 301, C.3.5; 1029, , 12, D.60; 354, 414, 474

Key Concept 2

Example 2 Explicit and Recursive Formulas Write an explicit formula and a recursive formula for finding the nth term in the sequence –1, 2, –4, …. First, find a common ratio. a 2 ÷ a 1 = 2 ÷ (–1) or –2Find the ratio between two pairs of consecutive terms to determine the common ratio. a 3 ÷ a 2 = –4 ÷ 2 or –2 The common ratio is –2.

Example 2 Explicit and Recursive Formulas Answer: a n = –1(–2) n –1 ; a 1 = –1; a n = (–2) a n – 1 For an explicit formula, substitute a 1 = –1 and r = –2 in the nth term formula. a n = a 1 r n – 1 nth term formula a n = –1(–2) n –1 a 1 = –1 and r = –2 The explicit formula is a n = –1(–2) n –1. For a recursive formula, state the first term a 1. Then indicate that the next term is the product of the previous term a n – 1 and r. a 1 = –1; a n = (–2) a n – 1 The recursive formula is a 1 = –1; a n = (–2) a n – 1.

Example 2 Find both an explicit formula and a recursive formula for the nth term of the geometric sequence 3, 16.5, 90.75, …. A.a n = 3(5.5) n – 1 ; a 1 = 3, a n = 5.5a n – 1 B.a n = 5.5(3) n – 1 ; a 1 = 3, a n = 5.5a n – 1 C.a n = 3(5.5) n – 1 ; a 1 = 3, a n = 5a n – 1 D.a n = – n ; a 1 = 3, a n = 5.5a n – 1

Example 3 nth Terms Find the 11th term of the geometric sequence –122, 115.9, – , …. First, find the common ratio. a 2 ÷ a 1 = ÷ (–122) or –0.95Find the ratio between two pairs of consecutive terms to determine the common ratio. a 3 ÷ a 2 = – ÷ or –0.95

Example 3 nth Terms Answer: about –73.05 Use the formula for the nth term of a geometric sequence. a n = a 1 r n – 1 nth term of a geometric sequence a 11 = –122(–0.95) 11 – 1 n = 11, a 1 = –122, and r = –0.95 a 11 ≈  Simplify. The 11th term of the sequence is about –73.05.

Example 3 Find the 25 th term of the geometric sequence 324, 291.6, , …. A.about B. about C. about D. about 28.72

Example 4 nth Term of a Geometric Sequence A. REAL ESTATE A couple purchased a home for $225,000. At the end of each year, the value of the home appreciates 3%. Write an explicit formula for finding the value of the home after n years. If the house’s value appreciates at a rate of 3% per year, it will have a value of 100% + 3% or 103% of its original value. Note that the original value given represents the a 0 and not the a 1 term, so we need to use an adjusted formula for the nth term of this geometric sequence.

Example 4 nth Term of a Geometric Sequence first terma 1 = a 0 r second terma 2 = a 0 r 2  nth terma n = a 0 r n Use this adjusted formula to find an explicit formula for the value of the house after n years. a n = a 0 r n Adjusted nth term of a geometric sequence a n = 225,000(1.03) n a 0 = 225,000, r = 1.03

Example 4 Answer:a n = 225,000(1.03) n nth Term of a Geometric Sequence An explicit formula for the value of the home after n years is a n = 225,000(1.03) n.

Example 4 nth Term of a Geometric Sequence B. REAL ESTATE A couple purchased a home for $225,000. At the end of each year, the value of the home appreciates 3%. What is the value of the home after the tenth year? Evaluate the formula you found in part A with n = 10. a n = 225,000(1.03) n Original formula = 225,000(1.03) 10 n = 10 ≈ 302,381.19Simplify.

Example 4 Answer:about $302, nth Term of a Geometric Sequence The value of the home after the tenth year is about $302, The value of the home at each year is shown as a point on the graph. The function connecting the points represents exponential growth.

Example 4 A. BOAT Jeremy purchased a boat for $12,500. By the end of each year, the value of the boat depreciates 4%. Write an explicit formula for finding the value of the boat after n years. A.a n = 12,500(1.04) n B.a n = 12,500(0.96) n C.a n = 12,500(0.96) n –1 D.a n = 12,000(0.96) n

Example 4 B. BOAT Jeremy purchased a boat for $12,500. At the end of each year, the value of the boat depreciates 4%. What is the value of the boat after 12 years? A.$20, B.$ C.$ D.$

Example 5 Geometric Means Write a sequence that has three geometric means between 264 and The sequence will resemble 264, _____, _____, _____, ?? ? Note that a 1 = 264, n = 5, and a 5 = Find the common ratio using the nth term for a geometric sequence formula. a 5 = a 1 r n – 1 nth term of a geometric sequence = 264r 5 –1 a 5 = , a 1 = 264, and n = 5

Example 5 Geometric Means = r 4 Simplify and divide each side by 264. ± = rTake the fourth root of each side. The common ratio is ±. Use r to find the geometric means.

Example 5 Answer:264, 66, 16.5, 4.125, or 264,  66, 16.5,  4.125, Geometric Means The sequence is 264, 66, 16.5, 4.125, and or 264,  66, 16.5,  4.125, and

Example 5 Write a geometric sequence that has two geometric means between 20 and A.20, 15, 11.25, B.20, 16.15, , C.20, , D.15, 11.25

Key Concept 6

Example 6 A. Find the sum of the first eleven terms of the geometric series 4, –6, 9, …. Sums of Geometric Series First, find the common ratio. a 2 ÷ a 1 = –6 ÷ 4 or –1.5 Find the ratio between two pairs of consecutive terms to determine the common ratio. a 3 ÷ a 2 = 9 ÷ (–6) or –1.5 The common ratio is –1.5. Use Formula 1 to find the sum of the series.

Example 6 Answer: about 140 Sums of Geometric Series Sum of a geometric series formula 1. The sum of the geometric series is about 140. n = 11, a 1 = 4, and r = –1.5 Simplify.

Example 6 B. Find the sum of the first n terms of a geometric series with a 1 = –4, a n = –65,536, and r = 2. Sums of Geometric Series Use Formula 2 for the sum of a finite geometric series. Simplify. a 1 = –4, a n = –65,536, and r = 2 Sum of a geometric series formula 2.

Example 6 Answer: –131,068 Sums of Geometric Series The sum of the first n terms of the geometric series is –131,068.

Example 6 Find the sum of the first 8 terms of the geometric series …. A.18, B.85, C.384, D.2,113,

Example 7 Geometric Sum in Sigma Notation Method 1Use Formula 1. Find n, a 1, and r. n = 8 – or 6Upper bound minus lower bound plus 1 a 1 = –2(–2) 3–1 or –8n = 3 r = –2 r is the base of the exponential function. Substitute n = 6, a 1 =  8, and r =  2 into Formula 1. Find.

Example 7 Geometric Sum in Sigma Notation Method 2Use Formula 2. Find a n. a n = a 1 r n  1 nth term of a geometric sequence Simplify. n = 6, a 1 = –8, and r = –2 Formula 1

Example 7 Geometric Sum in Sigma Notation = –8(–2) 6–1 a 1 =  8, r =  2, and n = 6 = 256 Simplify. Substitute a 1 =  8, a n = 256, and r =  2 into Formula 2. Formula 2 n = 6, a 1 =  8, a n = 256, and r =  2 Simplify. Therefore, = 168. Answer: 168

Example 7 A. 10,912 B. 32,768 C. 43,680 D. 174,752 Find.

Key Concept 8

Example 8 A. If possible, find the sum of the infinite geometric series …. First, find the common ratio. Sums of Infinite Geometric Series a 2 ÷ a 1 = 18 ÷ 24 or.Find the ratio between two pairs of consecutive terms to determine the common ratio. a 3 ÷ a 2 = 13.5 ÷ 18 or The common ratio r is, and. This infinite geometric series has a sum. Use the formula for the sum of an infinite geometric series.

Example 8 Answer:96 Sums of Infinite Geometric Series Sum of an infinite geometric series formula The sum of the infinite series is 96. Simplify. a 1 = 9 and r =

Example 8 B. If possible, find the sum of the infinite geometric series …. First, find the common ratio. a 2 ÷ a 1 = 0.66 ÷ 0.33 or 2 Find the ratio between two pairs of consecutive terms to determine the common ratio. a 3 ÷ a 2 = 1.32 ÷ 0.66 or 2 Sums of Infinite Geometric Series The common ratio r is 2, and |2| > 1. Therefore, this infinite geometric series has no sum. Answer:does not exist

Example 8 The common ratio r is 0.65, and |0.65| < 1. Therefore, this infinite geometric series has a sum. Find a 1. a 1 = 3(0.65) 2 – 1 Lower bound = 2 = 1.95 Use the formula for the sum of an infinite geometric series to find the sum. Sums of Infinite Geometric Series C. If possible, find.

Example 8 Sums of Infinite Geometric Series Answer: Sum of an infinite geometric series formula a 1 = 1.95 and r = 0.65 Simplify.

Example 8 A. 1.5 B. 7.5 C. 30 D. does not exist If possible, find the sum of the infinite geometric series.

Vocabulary geometric sequence common ratio geometric means geometric series