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Algebra1 Geometric Sequences

Algebra1 Geometric Sequences. Warm Up. Write a function to describe each of the following graphs. 1) The graph of f (x) = x 2 - 3 translated 7 units up. 2) The graph of f (x) = 2x 2 + 6 narrowed and translated 2 units down. 1) f (x) = x 2 + 4. 2) f (x) = 3x 2 + 4. Geometric Sequences.

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Algebra1 Geometric Sequences

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  1. Algebra1Geometric Sequences CONFIDENTIAL

  2. Warm Up Write a function to describe each of the following graphs. 1) The graph of f (x) = x2 - 3 translated 7 units up 2) The graph of f (x) = 2x2 + 6 narrowed and translated 2 units down 1) f (x) = x2 + 4 2) f (x) = 3x2 + 4 CONFIDENTIAL

  3. Geometric Sequences Bungee jumpers can use geometric sequences to calculate how high they will bounce. The table shows the heights of a bungee jumper’s bounces. The height of the bounces shown in the table above form a geometric sequence. In a geometric sequence , the ratio of successive terms is the same number r, called the common ratio. CONFIDENTIAL

  4. 1 3 9 27 Geometric Sequences Find the next three terms in each geometric sequence. A) 1, 3, 9, 27, … Step1: Find the value of r by dividing each term by the one before it. 3 = 3 9 = 3 27 = 3 1 3 9 The value of r is 3. CONFIDENTIAL

  5. 27 81 243 729 × 3 × 3 × 3 Step2: Multiply each term by 3 to find the next three terms. The next three terms are 81, 243, and 729. CONFIDENTIAL

  6. -16 4 -1 1 4 B) -16, 4, -1, 1 , … 4 Step1: Find the value of r by dividing each term by the one before it. 4 = -1 -1 = -11/4 = -1 -16 4 4 4 -1 4 The value of r is -1. 4 CONFIDENTIAL

  7. 1 -11 -1 4 16 64 256 × -1 4 × -1 4 × -1 4 Step2: Multiply each term by -1 to find the next 4 three terms. The next three terms are -1, 1, and 1. 16 64 256 CONFIDENTIAL

  8. Now you try! Find the next three terms in each geometric sequence. 1a) 5, -10, 20, -40, … 1b) 512, 384, 288, … 1a) 80, -160, 320 1b) 216, 162, 121.5 CONFIDENTIAL

  9. Geometric sequences can be thought of as functions. The term number, or position in the sequence, is the input of the function, and the term itself is the output of the function. To find the output an of a geometric sequence when n is a large number, you need an equation, or function rule. Look for a pattern to find a function rule for the sequence above. CONFIDENTIAL

  10. an = a1rn - 1 nth term 1st termCommon ratio The pattern in the table shows that to get the nth term, multiply the first term by the common ratio raised to the power n - 1. If the first term of a geometric sequence is a 1 , the nth term is a n , and the common ratio is r, then CONFIDENTIAL

  11. Finding the nth Term of a Geometric Sequence A) The first term of a geometric sequence is 128, and the common ratio is 0.5. What is the 10th term of the sequence? an = a1rn - 1 Write the formula. a10 = (128)(0.5)10 - 1 Substitute 128 for a1 , 10 for n, and 0.5 for r. a10 = (128)(0.5)9 Simplify the exponent. a10 = 0.25 Use a calculator. The 10th term of the sequence is 0.25. CONFIDENTIAL

  12. B) For a geometric sequence, a 1 = 8 and r = 3. Find the 5th term of this sequence. an = a1rn - 1 Write the formula. a5 = (8)(3)5 - 1 Substitute 8 for a1 , 5 for n, and 3 for r. a5 = (8)(3)4 Simplify the exponent. a5 = 648 Use a calculator. The 5th term of the sequence is 648. CONFIDENTIAL

  13. 8 -16 32 -64 C) What is the 13th term of the geometric sequence 8, -16, 32, -64, … ? Step1: Find the value of r by dividing each term by the one before it. -16 = -2 32 = -2 -64 = -2 8 -16 32 The value of r is -2. CONFIDENTIAL

  14. Step2: Plug the value of r in the following formula. an = a1rn - 1 Write the formula. a13 = (8)(-2)13 - 1 Substitute 8 for a1 , 13 for n, and -2 for r. a13 = (8)(-2)12 Simplify the exponent. a13 = 32,768 Use a calculator. The 13th term of the sequence is 32,768. CONFIDENTIAL

  15. Now you try! 2) What is the 8th term of the sequence 1000, 500, 250, 125, … ? 2) 7.8125 CONFIDENTIAL

  16. 200 80 32 80 = 0.4 32 = 0.4 200 80 Sports Application A bungee jumper jumps from a bridge. The diagram shows the bungee jumper’s height above the ground at the top of each bounce. The heights form a geometric sequence. What is the bungee jumper’s height at the top of the 5th bounce? The value of r is 0.4. CONFIDENTIAL

  17. an = a1rn - 1 Write the formula. a5 = (200)(0.4)5 - 1 Substitute 200 for a1 , 5 for n, and 0.4 for r. a5 = (200)(0.5)4 Simplify the exponent. a5 = 5.12 Use a calculator. The height of the 5th bounce is 5.12 feet. CONFIDENTIAL

  18. Now you try! 3) The table shows a car’s value for 3 years after it is purchased. The values form a geometric sequence. How much will the car be worth in the 10th year? 3) $1342.18 CONFIDENTIAL

  19. Assessment Find the next three terms in each geometric sequence. 1) 2, 4, 8, 16, … 2) 400, 200, 100, 50, … 3) 4, -12, 36, -108, … 4) -2, 10, -50, 250, … • 32, 64, 128 • 2) 25, 12.5, 6.25 • 3) 324, -972, 2916 • 4)-1250, 6250, -31,250 CONFIDENTIAL

  20. 5) The first term of a geometric sequence is 1, and the common ratio is 10. What is the 10th term of the sequence? 6) What is the 11th term of the geometric sequence 3, 6, 12, 24, … ? 5) 1,000,000,000 6) 3072 CONFIDENTIAL

  21. 7) In the NCAA men’s basketball tournament, 64 teams compete in round 1. Fewer teams remain in each following round, as shown in the graph, until all but one team have been eliminated. The numbers of teams in each round form a geometric sequence. How many teams compete in round 5? 7) 4 CONFIDENTIAL

  22. Find the missing term(s) in each geometric sequence. 8) 20, 40,___,____ , … 9) ___, 6, 18,___, … 8) 80, 160 9) 2, , , 54 CONFIDENTIAL

  23. Let’s review Geometric Sequences Bungee jumpers can use geometric sequences to calculate how high they will bounce. The table shows the heights of a bungee jumper’s bounces. The height of the bounces shown in the table above form a geometric sequence. In a geometric sequence , the ratio of successive terms is the same number r, called the common ratio. CONFIDENTIAL

  24. 1 3 9 27 Geometric Sequences Find the next three terms in each geometric sequence. A) 1, 3, 9, 27, … Step1: Find the value of r by dividing each term by the one before it. 3 = 3 9 = 3 27 = 3 1 3 9 The value of r is 3. CONFIDENTIAL

  25. 27 81 243 729 × 3 × 3 × 3 Step2: Multiply each term by 3 to find the next three terms. The next three terms are 81, 243, and 729. CONFIDENTIAL

  26. Geometric sequences can be thought of as functions. The term number, or position in the sequence, is the input of the function, and the term itself is the output of the function. To find the output an of a geometric sequence when n is a large number, you need an equation, or function rule. Look for a pattern to find a function rule for the sequence above. CONFIDENTIAL

  27. an = a1rn - 1 nth term 1st termCommon ratio The pattern in the table shows that to get the nth term, multiply the first term by the common ratio raised to the power n - 1. If the first term of a geometric sequence is a 1 , the nth term is a n , and the common ratio is r, then CONFIDENTIAL

  28. Finding the nth Term of a Geometric Sequence A) The first term of a geometric sequence is 128, and the common ratio is 0.5. What is the 10th term of the sequence? an = a1rn - 1 Write the formula. a10 = (128)(0.5)10 - 1 Substitute 128 for a1 , 10 for n, and 0.5 for r. a10 = (128)(0.5)9 Simplify the exponent. a10 = 0.25 Use a calculator. The 10th term of the sequence is 0.25. CONFIDENTIAL

  29. 200 80 32 80 = 0.4 32 = 0.4 200 80 Sports Application A bungee jumper jumps from a bridge. The diagram shows the bungee jumper’s height above the ground at the top of each bounce. The heights form a geometric sequence. What is the bungee jumper’s height at the top of the 5th bounce? The value of r is 0.4. CONFIDENTIAL

  30. an = a1rn - 1 Write the formula. a5 = (200)(0.4)5 - 1 Substitute 200 for a1 , 5 for n, and 0.4 for r. a5 = (200)(0.5)4 Simplify the exponent. a5 = 5.12 Use a calculator. The height of the 5th bounce is 5.12 feet. CONFIDENTIAL

  31. You did a great job today! CONFIDENTIAL

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