7.5 Use Recursive Rules with 7.5 Use Recursive Rules with Sequences and FunctionsSequences and Functions

p. 467p. 467

• What is a recursive rule for arithmetic sequences?

• What is a recursive rule for geometric sequences?

• What is an iteration?

Explicit RuleExplicit Rule

Recursive RuleRecursive Rule

• Gives the beginning term(s) of a sequence and a recursive rule that relates the given term(s) to the next terms in the sequence.

• For example: Given a0=1 and an=an-1-2

• The 1st five terms of this sequence would be: a0, a1, a2, a3, a4 OR

• 1, -1, -3, -5, -7

Recursive Equations

Example: Write the indicated rule for the Example: Write the indicated rule for the arithmetic sequence with aarithmetic sequence with a11=15 and d=5.=15 and d=5.

• Explicit rule

an=a1+(n-1)d

an=15+(n-1)5

an=15+5n-5

an=10+5n

• Recursive rule

(*Use the idea that you get the next term by adding 5 to the previous term.)

Or an=an-1+5

So, a recursive rule would be a1=15, an=an-1+5

Example: Write the indicated rule for the Example: Write the indicated rule for the geometric sequence with ageometric sequence with a11=4 and r=0.2.=4 and r=0.2.

• Explicit rule

an=a1rn-1

an=4(0.2)n-1

• Recursive rule

(*Use the idea that you get the next term by multiplying the previous

term by 0.2)

Or an=r*an-1=0.2an-1

So, a recursive rule for the sequence would be a1=4, an=0.2an-1

Example: Write the 1Example: Write the 1stst 5 terms of 5 terms of the sequence.the sequence.

• a1=2, a2=2, an=an-2-an-1

a3=a3-2-a3-1→a1-a2=2-2=0

a4=a4-2-a4-1→a2-a3=2-0=2

a5=a5-2-a5-1→a3-a4=0-2=-2

2, 2, 0, 2, -2

11stst term term 22ndnd term term 1

2

3

4

5

2

2

0

2

-2

Write the first six terms of the sequence.Write the first six terms of the sequence.a. a. aa00 = 1, = 1, aann = = aan n – 1– 1 + 4 + 4 b. b. aa11 = 1, = 1, aann = 3= 3aan n – 1– 1

SOLUTIONSOLUTION

a. a. aa00 = 1 = 1

aa11 = = aa00 + 4 = 1 + 4 = 5 + 4 = 1 + 4 = 5

aa22 = = aa11 + 4 = 5 + 4 = 9 + 4 = 5 + 4 = 9

aa33 = = aa22 + 4 = 9 + 4 = 13 + 4 = 9 + 4 = 13

aa44 = = aa33 + 4 = 13 + 4 = 17 + 4 = 13 + 4 = 17

aa55 = = aa44 + 4 = 17 + 4 = 21 + 4 = 17 + 4 = 21

b. b. aa11 = 1 = 1

aa22 = 3 = 3aa11 = 3(1) = 3 = 3(1) = 3

aa33 = 3 = 3aa22 = 3(3) = 9 = 3(3) = 9

aa44 = 3 = 3aa33 = 3(9) = 27 = 3(9) = 27

aa55 = 3 = 3aa44 = 3(27) = 81 = 3(27) = 81

aa66 = 3 = 3aa55 = 3(81) = 243 = 3(81) = 243

a. a. 3, 13, 23, 33, 43, . . .3, 13, 23, 33, 43, . . .

SOLUTIONSOLUTION

The sequence is arithmetic with first term The sequence is arithmetic with first term aa11 = =

33 and common difference and common difference d d = 13 – 3 = 10= 13 – 3 = 10..aann = = aan n –– 1 1 + + dd= = aan n – – 11 + + 1010

General recursive equation for General recursive equation for aann

Substitute Substitute 1010 for for dd..

ANSWERANSWER

So, a recursive rule for the sequence isSo, a recursive rule for the sequence is aa11 = 3, = 3, aann = = aann – 1 – 1 + 10+ 10..

Write the first six terms of the sequence.Write the first six terms of the sequence.

b. The sequence is geometric with first term b. The sequence is geometric with first term aa11 = 16 = 16 and common ratio and common ratio r r == 4040

1616 = 2.5.= 2.5.

aann = = rr aann – 1– 1

= = 2.52.5aan n – 1– 1

General recursive equation for General recursive equation for aann

Substitute Substitute 2.52.5 for for rr..

So, a recursive rule for the sequence is So, a recursive rule for the sequence is aa11 = 16, = 16, aann = 2.5= 2.5aan n – 1– 1..

ANSWERANSWER

Write the first six terms of the sequence.Write the first six terms of the sequence.

b. b. 16, 40, 100, 250, 16, 40, 100, 250, 625, . . .625, . . .

Write the first five terms of the sequence.Write the first five terms of the sequence.1. 1. aa11 = 3, = 3, aann = = aan n – 1– 1 7 7––SOLUTIONSOLUTION

aa11 = 3 = 3

aa33 = = aa22 –– 7 7 aa33 = = –– 4 4 –– 7 = 7 = –– 11 11aa44 = = aa33 –– 7 = 7 = –– 11 11 –– 7 = 7 = –– 18 18aa55 = = aa44 –– 7 = 7 = –– 18 18 –– 7 = 7 = –– 25 25

aa22 = = aa11 7 = 3 7 = 4 7 = 3 7 = 4–– ––––

ANSWERANSWER 3, –4, –11, –18, –253, –4, –11, –18, –25

33−−44

−−1111−−1818−−2525

Or think of it this way…Or think of it this way…

Write the first five terms of the sequence.Write the first five terms of the sequence.3. 3. aa00 = 1, = 1, aann = = aan n – 1– 1 + + nnSOLUTIONSOLUTION

aa00 = 1 = 1

aa11 = = aa00 ++ 1 = 1 1 = 1 ++ 1 = 1 = 22

aa2 2 = = aa11 ++ 1 = 1 = 22 ++ 2 = 4 2 = 4

aa33 = = aa22 ++ 3 = 4 3 = 4 ++ 3 = 7 3 = 7

aa44 = = aa33 ++ 4 = 7 4 = 7 ++ 4 = 11 4 = 11

ANSWERANSWER 1, 2, 4, 7, 111, 2, 4, 7, 11

Write the first five terms of the sequence.Write the first five terms of the sequence.4. 4. aa11 = 4, = 4, aann = 2= 2aan n – 1– 1 –– 1 1

SOLUTIONSOLUTION

aa11 = 4 = 4aa22 = 2 = 2aa11 –– 1 = (2 4) 1 = (2 4) –– 1 = 8 1 = 8 –– 1 = 7 1 = 7

aa33 = 2 = 2aa22 –– 1 = (2 7) 1 = (2 7) –– 1 = 14 1 = 14 –– 1 = 13 1 = 13

aa55 = 2 = 2aa44 –– 1 = (2 25) 1 = (2 25) –– 1 = 49 1 = 49

aa44 = 2 = 2aa33 –– 1 = (2 13) 1 = (2 13) –– 1 = 26 1 = 26 –– 1 = 25 1 = 25

ANSWERANSWER 4, 7, 13 25, 494, 7, 13 25, 49

Write a recursive rule for the sequence.Write a recursive rule for the sequence.5. 5. 2, 14, 98, 686, 4802, . . .2, 14, 98, 686, 4802, . . .

The sequence is geometric with first term The sequence is geometric with first term aa11

= 2= 2 and common ratio and common ratior r ==

aa22

aa11= 7= 7

aann = = rr aann – 1– 1

= = 7 7 ·· aan n – 1– 1

SOLUTIONSOLUTION

So, a recursive rule for the sequence isSo, a recursive rule for the sequence isANSWERANSWER

aa11 = 2 = 2, , aann = 7= 7aan n – 1– 1

Write a recursive rule for the sequence.Write a recursive rule for the sequence.

a. a. 1, 1, 2, 3, 5, . . .1, 1, 2, 3, 5, . . .

SOLUTIONSOLUTION

Beginning with the third term in the Beginning with the third term in the sequence, each term is the sum of the two sequence, each term is the sum of the two previous terms.previous terms.

a.a.

ANSWERANSWER

So, a recursive rule is So, a recursive rule is aa11 = 1, = 1, aa22 = 1, = 1, aann = = aann – 2 – 2 + + aann –– 11..

This sequence is the Fibonacci sequence.This sequence is the Fibonacci sequence.By definition, the first two numbers in the Fibonacci By definition, the first two numbers in the Fibonacci sequence are 0 and 1 (alternatively, 1 and 1), and each sequence are 0 and 1 (alternatively, 1 and 1), and each subsequent number is the sum of the previous two.subsequent number is the sum of the previous two.0,1,1,2,3,5,8,13,21,34,55,89,144,…0,1,1,2,3,5,8,13,21,34,55,89,144,…

Write a recursive rule for the sequence.Write a recursive rule for the sequence.

b. b. 1, 1, 2, 6, 24, . . .1, 1, 2, 6, 24, . . .

SOLUTIONSOLUTION

So, a recursive rule isSo, a recursive rule is aa00 = 1, = 1, aann = = n n aan –n – 1 1..

ANSWERANSWER

b. b. Denote the first term by Denote the first term by aa0 0 = 1= 1. Then note that . Then note that

aa11 = 1= 1 = = 11 aa00, , aa22 = 2 = = 2 = 22 aa11, , aa33 = 6 = = 6 = 33 aa22, , and and

so on.so on.

This sequence lists factorial numbers.This sequence lists factorial numbers.

Iterating FunctionsIterating Functions

Find the first three iterates Find the first three iterates xx11, , xx22, and , and xx33 of the of the

function function f f ((xx) = –3) = –3x +x + 1 1 for an initial value of for an initial value of xx0 0 = 2= 2..

SOLUTIONSOLUTION

xx11 = = f f ((xx00))

= = f f ((22))= –3(= –3(22) + 1) + 1= = – – 55

xx22 = = f f ((xx11))= = f f ((–5–5))= –3(= –3(2255) + 1) + 1= = 1616

xx33 = = f f ((xx22))= = f f ((1616))= –3(= –3(1616) + 1) + 1= – 47= – 47

The first three iterates areThe first three iterates are – 5, 16,– 5, 16, and and – 47.– 47.

ANSWERANSWER

Find the first three iterates of the function for Find the first three iterates of the function for the initial value.the initial value.

11.11. ff ( (xx) = 4) = 4xx – 3, – 3, xx0 0 == 22

SOLUTIONSOLUTION

xx11 = = f f ((xx00))

= = f f ((22))= 8 – 3= 8 – 3== 55

xx22 = = f f ((xx11))= 4 (= 4 (55) – 3) – 3= = 1717

xx33 = = f f ((xx22))

= 4= 4 ((1717) – 3 ) – 3 = 68 – 3= 68 – 3= 65= 65

The first three iterates areThe first three iterates are 5, 17,5, 17, and and 65.65.

ANSWERANSWER

7.5 Assignment:7.5 Assignment:

p. 470, 3-27 odd, skip 21. 470, 3-27 odd, skip 21

Write a recursive rule for the Write a recursive rule for the sequence 1,2,2,4,8,32,… .sequence 1,2,2,4,8,32,… .

• First, notice the sequence is neither arithmetic nor geometric.

• So, try to find the pattern.

• Notice each term is the product of the previous 2 terms.

• Or, an-1*an-2

• So, a recursive rule would be:

a1=1, a2=2, an= an-1*an-2

Example: Write a recursive rule for Example: Write a recursive rule for the sequence 1,1,4,10,28,76.the sequence 1,1,4,10,28,76.

• Is the sequence arithmetic, geometric, or neither?

• Find the pattern.

• 2 times the sum of the previous 2 terms

• Or 2(an-1+an-2)

• So the recursive rule would be:

a1=1, a2=1, an= 2(an-1+an-2)