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**Algebra1 Geometric Sequences**

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**Write a function to describe each of the following graphs.**

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

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**The table shows the heights of a bungee jumper’s bounces.**

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

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**Find the next three terms in each geometric sequence.**

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 The value of r is 3. CONFIDENTIAL

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**The next three terms are 81, 243, and 729.**

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

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**Step1: Find the value of r by dividing each term by the one before it.**

4 Step1: Find the value of r by dividing each term by the one before it. 4 4 = = /4 = -1 The value of r is -1. 4 CONFIDENTIAL

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**Step2: Multiply each term by -1 to find the next 4 three terms.**

× -1 4 The next three terms are -1, 1, and 1. CONFIDENTIAL

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**Find the next three terms in each geometric sequence.**

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

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**Geometric sequences can be thought of as functions**

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

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**nth term 1st term Common 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 nth term 1st term Common ratio an = a1rn - 1 CONFIDENTIAL

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**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

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**The 5th term of the sequence is 648.**

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

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**Step1: Find the value of r by dividing each term by the one before it.**

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 The value of r is -2. CONFIDENTIAL

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**The 13th term of the sequence is 32,768.**

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

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**2) What is the 8th term of the sequence 1000, 500, 250, 125, … ?**

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

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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? 80 = = 0.4 The value of r is 0.4. CONFIDENTIAL

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**The height of the 5th bounce is 5.12 feet.**

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

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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) $ CONFIDENTIAL

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**Find the next three terms in each geometric sequence.**

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

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**6) What is the 11th term of the geometric sequence 3, 6, 12, 24, … ?**

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

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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

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**Find the missing term(s) in each geometric sequence.**

8) 20, 40,___,____ , … 9) ___, 6, 18,___, … 8) 80, 160 9) 2, , , 54 CONFIDENTIAL

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**The table shows the heights of a bungee jumper’s bounces.**

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

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**Find the next three terms in each geometric sequence.**

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 The value of r is 3. CONFIDENTIAL

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**The next three terms are 81, 243, and 729.**

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

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**Geometric sequences can be thought of as functions**

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

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**nth term 1st term Common 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 nth term 1st term Common ratio an = a1rn - 1 CONFIDENTIAL

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**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

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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? 80 = = 0.4 The value of r is 0.4. CONFIDENTIAL

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**The height of the 5th bounce is 5.12 feet.**

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

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**You did a great job today!**

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