7-7: Geometric Sequences as Exponential Functions

1. 7-7: Geometric Sequences as Exponential Functions Essential Skills: Identify and generate geometric sequences Relate geometric sequences to exponential functions

2. 7-7: Geometric Sequences • Geometric Sequence: A pattern where the first term is not zero, and each term after the first can be found by multiplying the previous term by a common ratio. • Common ratio: The number that multiplies each term in a geometric sequence.

3. 7-7: Geometric Sequences • Example 1A: Determine whether the sequence is arithmetic, geometric, or neither. Explain. • 0, 8, 16, 24, 32, … • Try subtracting consecutive numbers to see if the pattern is algebraic • 8 – 0 = 8 • 16 – 8 = 8 • 24 – 16 = 8 • 32 – 24 = 8 • The common difference is 8, so the sequence is algebraic

4. 7-7: Geometric Sequences • Example 1B: Determine whether the sequence is arithmetic, geometric, or neither. Explain. • 64, 48, 36, 27, … • Try subtracting consecutive numbers to see if the pattern is algebraic • 48 – 64 = -16 • 36 – 48 = -12 It’s not algebraic • Try dividing consecutive numbers to see if the pattern is geometric • 48/64 = 3/4 • 36/48 = 3/4 • 27/36 = 3/4 • The common ratio is ¾, so the sequence is geometric

5. 1) The pattern 1, 7, 49, 343, … is • Arithmetic • Geometric • Neither

6. 2) The pattern 1, 2, 4, 14, 54, … is • Arithmetic • Geometric • Neither

7. 7-7: Geometric Sequences • Example 2A: Find the next three terms in the geometric sequence • 1, -8, 64, -512, … • Step 1: Find the common ratio • -8/1 = -8 • 64/-8 = -8 • -512/64 = -8 • The common ratio is -8 • Step 2: Continue the pattern with the common ratio • -512 ● -8 = 4096 • 4096 ● -8 = -32,768 • -32,768 ● -8 = 262,144 • The next three terms are 4096, -32,768, and 262,144

8. 7-7: Geometric Sequences • Example 2B: Find the next three terms in the geometric sequence • 40, 20, 10, 5, … • Step 1: Find the common ratio • 20/40 = ½ • 10/20 = ½ • 5/10 = ½ • The common ratio is ½ • Step 2: Continue the pattern with the common ratio • 5 ● ½ = 5/2 • 5/2● ½ = 5/4 • 5/4● ½ = 5/8 • The next three terms are 5/2, 5/4, and 5/8

9. 3) Find the next three terms in the geometric sequence: 1, -5, 25, -125 • 250, -500, 1000 • 150, -175, 200 • -250, 500, -1000 • 625, -3125, 15625

10. 4) Find the next three terms in the geometric sequence: 800, 200, 50, 25/2 • 15, 10, 5 • 25/8, 25/32, 25/128 • 12, 3, ¾ • 0, -25, -50

11. 7-7: Geometric Sequences • nth term of a Geometric Sequence • We can find any term in a geometric sequence if we know two things: • The first term in the sequence, called a1 • The common ratio r • The nth term in a sequence can then be found using the formula • an = a1rn-1 • Hint: Do the “n-1” exponent first, and

12. 7-7: Geometric Sequences • Example 3A: Write an equation for the nth term of a geometric sequence: 1, -2, 4, -8, … • Step 1: Find the first term • 1 • Step 2: Find the common ratio • r = -2/1 = -2 • Step 3: Substitute into the equation an = a1rn-1 • an = 1(-2)n-1 • Example 3B: Find the 12th term of the sequence • Substitute 12 for n and solve • a12 = 1(-2)12-1 • a12 = 1(-2)11 • a12 = -2048

13. 5) Write an equation for the nth term of the geometric sequence: 3, -12, 48, -192 • an = 3(-4)n-1 • an = 3(¼)n-1 • an = 3(1/3) n-1 • an = 4(-3)n-1

14. 6) Find the 7th term of this sequence using the equation an = 3(-4)n-1 • 768 • -3072 • 12,288 • -49,152

15. Leader Board (6 points)

16. 7-7: Geometric Sequences • Assignment • Page 441 • 1 – 11, 15 – 29 (odds)