7-8: RECURSIVE FORMULASEssential Skills:Use a recursive formula to list terms in a sequenceWrite recursive formulas for arithmetic and geometric sequences

7-8: Recursive Formulas Example 1

Find the first five terms of the sequence in which a1 = -8 and an = -2an-1 + 5 if n > 2 an-1 means the previous number in the sequence a2 = -2a1 + 5

= -2(-8) + 5= 21

a3 = -2a2 + 5= -2(21) + 5= -37

7-8: Recursive Formulas Example 1

Find the first five terms of the sequence in which a1 = -8 and an = -2an-1 + 5 if n > 2 a2 = 21 a3 = -37 a4 = -2a3 + 5

= -2(-37) + 5= 79

a5 = -2a4 + 5= -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 a1 = -3 and an = 4an-1 – 9 if n > 2

1 2 3 4

6%

41%

12%

41%1. -3, -12, -48, -192, -7682. -3, -21, -93, -381, -15333. -12, -48, -192, -768, -

30724. -21, -93, -381, -1533, -

6141

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21 22 23

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: an = an-1 + d, where d is

the common ratio Geometric Sequence: an = r ● an-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 = 6 35 – 29 = 6 41 – 35 = 6 This is arithmetic with a common difference of 6

Step 2: Use the formula for an arithmetic sequence an = an-1 + 6

Step 3: The first term is 23 Recursive Formula: a1 = 23 and an = an-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 = -28 63 – (-21) = 84 not arithmetic Divide consecutive terms to see if it’s geometric -21/7 = -3 63/-21 = -3 -189/63 = -3 The sequence is geometric with common ratio of -3

Step 2: Use the formula for an arithmetic sequence an = -3an-1

Step 3: The first term is 7 Recursive Formula: a1 = 7 and an = -3an-1

2) Write a recursive formula for -3, -12, -21, -30, …

1 2 3 4

0%

13%

56%

31%

1. a1 = -3, an = -4an-1

2. a1 = -3, an = 4an-1

3. a1 = -3, an = an-1 – 9

4. a1 = -3, an = an-1 + 9

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21 22 23 24

Assignment Page 448 – 449

Problems 1 – 3 & 11 – 21 (odds)

7-8: RECURSIVE FORMULASDAY 2Essential Skills:Use a recursive formula to list terms in a sequenceWrite 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 7200/12000 = 3/5 4320/7200 = 3/5 2592/4320 = 3/5

Step 2: Use the formula for a geometric sequence an = r ● an-1 an = 3/5 an-1 and a1 = 12,000

Year

Price ($)

1 12,0002 72003 43204 2592

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 an = a1rn-1

an = 12000(3/5)n-1

Year

Price ($)

1 12,0002 72003 43204 2592

3) The value of a home has increased each year. Write a recursive and explicit formula for the sequence.

1 2 3 4

50%

0%

13%

38%

Year

Value ($)

1 157,000

2 160,500

3 164,000

4 167,500

1. a1 = 157,000, an = an-1 + 3500an = 157,000 + 3500n

2. a1 = 157,000, an = an-1 + 3500an = 153,500 + 3500n

3. a1 = 153,500, an = an-1 + 3500an = 153,500 + 3500n

4. a1 = 153,500, an = an-1 + 3500an = 157,000 + 3500n

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7-8: Recursive Formulas Example 4A: Write a recursive form for an =

2n – 4 an = 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 a1 a1 = 2(1) – 4 a1 = -2

a1 = -2 and an = an-1 + 2

7-8: Recursive Formulas Example 4B: Write an explicit form for a1

= 84 and an = 1.5an-1 Step 1: Find a1

Oh, wait… it was given to you Step 2: Find r (since this is geometric)

Explicit Form is an = r ● an-1 So the number in front of the an-1 is the

common ratio an = 84(1.5)n-1

4) Write an explicit formula for a1 = 9 and an = 0.2an-1

1 2 3 4

0%

92%

0%8%

an = 45(0.2)n-1

an = 9(0.2)n+1

an = 9(0.2)n

an = 9(0.2)n-1

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7-8: Recursive Functions Assignment

Page 448 5 – 9, 23 – 27 (odds)