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Warm Up Section 3.6B (1). Show that f(x) = 3x + 5 and g(x) = are inverses. (2). Find the inverse of h(x) = 8 – 3x. (3). Solve: 27 x – 1 < 9 2x + 3 (4). Is k(x) = 4.6 x increasing or decreasing throughout its domain? x – 5 3

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Answers to Warm Up Section 3.6B (1). f(g(x)) g(f(x)) = f( ) = g(3x + 5) = 3( ) + 5 = = x – 5 + 5 = = x = x x – 5 3 x – 5 3 3x + 5 – 5 3 3x 3

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(2). h -1 (x) = (3). x > -9 (4). k(x) = 4.6 x is increasing throughout its domain x – 8 -3

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Geometric Sequences and Series Section 3.6 Standard: MM2A2 f g Essential Question: What are the sums of finite geometric sequences and series? How do geometric sequences relate to exponential functions?

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Let’s begin our lesson with a graphing review of exponential functions.

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1) Graph y = 3 x-1 +2. -8 -6 -4 -2 2 4 6 8 86428642 -2 -4 -6 -8 xy Growth xy

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Domain: Range: Asymptotes: Zeros: y-intercept: Interval of increasing: Interval of decreasing: Rate of change (2 ≤ x ≤ 3) End behavior: All reals y > 2 y = 2 None (0, 2.33) All reals None (2, 5) (3, 11) x → -∞, y → 2; x → ∞, y → ∞

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2) Graph -8 -6 -4 -2 2 4 6 8 86428642 -2 -4 -6 -8 xy Decay

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Domain: Range: Asymptotes: Zeros: Y-intercept: Interval of increasing: Interval of decreasing: Rate of change (-3 ≤ x ≤ -2) End behavior: All reals y > -4 y = -4 (-5, 0) (0, -.3875) All reals None (-3, -3) (-2, -3.5) x → -∞, y → ∞; x → ∞, y → -4

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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. Geometric Series: The expression formed by adding the terms of a geometric sequence.

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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: a n = a 1 + d(n – 1).

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1.Identify the next three terms of the arithmetic sequence, then write the explicit formula for the sequence: 3, 7, 11, 15, a n = 3 + 4(n – 1) or a n = 4n – 1 2.Use the formula from example #1 to find the 27 th term of the sequence. a 27 = 3 + 4(27 – 1) = 3.What is the sum of the first 27 terms of this sequence. Hint: use 19, 23, 27,... 107

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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. 4. 2, 6, 18, 54, 162,... 5. 5, 50, 500, 5000,... 6. 3,,,,... 7. -4, 24, -144, 864, -5184,... r = 3 r = 10 r = ½ r = -6

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Tell whether the sequences is arithmetic, geometric or neither. For arithmetic sequences, give the common difference. For geometric sequences, give the common ratio. 8. 5, 10, 15, 20, 25, …. 9. 1, 1, 2, 3, 5, 8, 13, 21, … 10. 1, -4, 16, -64, 256, … 11. 512, 256, 128, 64, 32, … arithmetic; d = 5 neither geometric; r = -4 geometric; r = ½

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Check for Understanding: 12. Find the first four terms of a geometric sequence in which a 1 = 5 and r = -3. _____, _____, _____, _____. 13. Find the missing term in the geometric sequence: -7, _______, -28, 56, _______,... 5-1545-135 × -3 56 ÷ -28 = -2 So, r = -2 × -2 14 × -2 -112

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Investigation 2: The explicit formula used to find the n th term of a geometric sequence with the first term a 1 and the common ratio r is given by: a n = a 1 ∙ r n-1 Write a rule for the n th term of the sequence given. Then find a 10. 14. 972, -324, 108, -36, … Rule: a n = 972∙(-⅓) n-1 a 10 = 972∙(-⅓) 10-1 = ____

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15. 1, 6, 36, 216, 1296, … Rule: a n = 1∙6 n-1 a 10 = 1∙6 10-1 = 10077696 16.14, 28, 56, 112, … Rule: a n = 14∙2 n-1 a 10 = 14∙2 10-1 = 7168

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Check for Understanding: 17. If a 5 = 324 and r = -3, write the explicit formula for the geometric sequence and find a 10. _____, _____, _____, _____, 324 Rule: a n = 4∙(-3) n-1 a 10 = 4∙(-3) 10-1 = -78732 ÷ -3 -108 ÷ -3 36 ÷ -3 -12 ÷ -3 4 OR

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

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19. If a 3 = 56 and a 6 = 448 complete the following for the geometric sequence: _____, _____, 56, _____, _____, 448 ÷ 2 28 14 × 2 112 224

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20. If r = 2 and a 1 = 1 for a geometric sequence, a.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) c.What kind of graph does this represent? exponential

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