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

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.

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

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 = – 48 = -12It’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

1) The pattern 1, 7, 49, 343, … is 1. Arithmetic 2. Geometric 3. Neither

2) The pattern 1, 2, 4, 14, 54, … is 1. Arithmetic 2. Geometric 3. Neither

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 The common ratio is -8 Step 2: Continue the pattern with the common ratio -512 ● -8 = ● -8 = -32, ,768 ● -8 = 262,144 The next three terms are 4096, -32,768, and 262,144

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

3) Find the next three terms in the geometric sequence: 1, -5, 25, , -500, , -175, , 500, , -3125, 15625

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

7-7: Geometric Sequences n th 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 a 1 The common ratio r The nth term in a sequence can then be found using the formula a n = a 1 r n-1 Hint: Do the “n-1” exponent first, and

7-7: Geometric Sequences Example 3A: Write an equation for the n th 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 a n = a 1 r n-1 a n = 1(-2) n-1 Example 3B: Find the 12 th term of the sequence Substitute 12 for n and solve a 12 = 1(-2) 12-1 a 12 = 1(-2) 11 a 12 = -2048

5) Write an equation for the n th term of the geometric sequence: 3, -12, 48, a n = 3(-4) n-1 2. a n = 3(¼) n-1 3. a n = 3( 1 / 3 ) n-1 4. a n = 4(-3) n-1

6) Find the 7 th term of this sequence using the equation a n = 3(-4) n , ,152

Leader Board (6 points) 6Jemikah Harrell 6Madison Roney 6Daniel Helper 6Andrew Duncan 6Isaiah Hanlan

7-7: Geometric Sequences Assignment Page – 11, 15 – 29 (odds)