7-8: RECURSIVE FORMULAS Essential Skills: Use a recursive formula to list terms in a sequence Write recursive formulas for arithmetic and geometric sequences

7-8: Recursive Formulas  Example 1  Find the first five terms of the sequence in which a 1 = -8 and a n = -2a n if n > 2 a n-1 means the previous number in the sequence a 2 = -2a = -2(-8) + 5 = 21 a 3 = -2a = -2(21) + 5 = -37

7-8: Recursive Formulas  Example 1  Find the first five terms of the sequence in which a 1 = -8 and a n = -2a n if n > 2 a 2 = 21a 3 = -37 a 4 = -2a = -2(-37) + 5 = 79 a 5 = -2a = -2(79) + 5 = -153 The first five terms are: -8, 21, -37, 79, -153

1) Find the first five terms of the sequence in which a 1 = -3 and a n = 4a n-1 – 9 if n > , -12, -48, -192, , -21, -93, -381, , -48, -192, -768, , -93, -381, -1533, -6141

7-6: Recursive Functions  Writing Recursive Functions  Step 1: Determine if the sequence is arithmetic or geometric by finding the common difference or common ratio.  Step 2: Write a recursive formula Arithmetic Sequence: a n = a n-1 + d, where d is the common ratio Geometric Sequence: a n = r ● a n-1 where r is the common ratio  Step 3: State the first term for n.

7-6: Recursive Functions  Example 2A: Write a recursive formula for the sequence 23, 29, 35, 41, …  Step 1: Is it arithmetic or geometric Subtract consecutive terms to see if it’s arithmetic 29 – 23 = 635 – 29 = 641 – 35 = 6 This is arithmetic with a common difference of 6  Step 2: Use the formula for an arithmetic sequence a n = a n  Step 3: The first term is 23  Recursive Formula: a 1 = 23 and a n = a n-1 + 6

7-6: Recursive Functions  Example 2B: Write a recursive formula for the sequence 7, -21, 63, -189, …  Step 1: Is it arithmetic or geometric? Subtract consecutive terms to see if it’s arithmetic -21 – 7 = – (-21) = 84not arithmetic Divide consecutive terms to see if it’s geometric -21 / 7 = / -21 = / 63 = -3 The sequence is geometric with common ratio of -3  Step 2: Use the formula for an arithmetic sequence a n = -3a n-1  Step 3: The first term is 7  Recursive Formula: a 1 = 7 and a n = -3a n-1

2) Write a recursive formula for -3, -12, -21, -30, … 1. a 1 = -3, a n = -4a n-1 2. a 1 = -3, a n = 4a n-1 3. a 1 = -3, a n = a n-1 – 9 4. a 1 = -3, a n = a n-1 + 9

Assignment  Page 448 – 449  Problems 1 – 3 & 11 – 21 (odds)

7-8: RECURSIVE FORMULAS DAY 2 Essential Skills: Use a recursive formula to list terms in a sequence Write recursive formulas for arithmetic and geometric sequences

7-8: Recursive Formulas  Example 3A  The price of a car depreciates at the end of each year. Write a recursive formula for the sequence  Step 1: Find the common ratio 7200 / = 3 / / 7200 = 3 / / 4320 = 3 / 5  Step 2: Use the formula for a geometric sequence a n = r ● a n-1 a n = 3 / 5 a n-1 and a 1 = 12,000 YearPrice ($) 112,

7-8: Recursive Formulas  Example 3B  Write an explicit formula for the sequence  Step 1: Find the common ratio 3 / 5  Step 2: Use the formula for a geometric sequence a n = a 1 r n-1 a n = 12000( 3 / 5 ) n-1 YearPrice ($) 112,

3) The value of a home has increased each year. Write a recursive and explicit formula for the sequence. YearValue ($) 1157, , , , a 1 = 157,000, a n = a n a n = 157, n 2. a 1 = 157,000, a n = a n a n = 153, n 3. a 1 = 153,500, a n = a n a n = 153, n 4. a 1 = 153,500, a n = a n a n = 157, n

7-8: Recursive Formulas  Example 4A: Write a recursive form for a n = 2n – 4  a n = 2n – 4  Step 1: Determine if arithmetic/geometric Since we’re subtracting a term, this is arithmetic. The common difference is 2  Step 2: Find a 1 a 1 = 2(1) – 4 a 1 = -2  a 1 = -2 and a n = a n-1 + 2

7-8: Recursive Formulas  Example 4B: Write an explicit form for a 1 = 84 and a n = 1.5a n-1  Step 1: Find a 1 Oh, wait… it was given to you  Step 2: Find r (since this is geometric) Explicit Form is a n = r ● a n-1 So the number in front of the a n-1 is the common ratio  a n = 84(1.5) n-1

4) Write an explicit formula for a 1 = 9 and a n = 0.2a n-1  a n = 45(0.2) n-1  a n = 9(0.2) n+1  a n = 9(0.2) n  a n = 9(0.2) n-1

7-8: Recursive Functions  Assignment  Page 448  5 – 9, 23 – 27 (odds)