6.4 Recursion Formulas
6.4 Recursion Formulas In previous sections, you have determined sequences using a formula for the nth term. An example is the formula tn = 2n + 3 or f(n) = 2n + 3, which determines the arithmetic sequence 5, 7, 9, 11, 13, … Another example is the formula n−1 or f(n) = 2n − 1, which tn = 2 determines the geometric sequence 1, 2, 4, 8, 16, … Such formulas are known as explicit formulas. They can be used to calculate any term in a sequence without knowing the previous term. For example, the tenth term in the sequence determined by the formula tn = 2n + 3 is 2(10) + 3, or 23. It is sometimes more convenient to calculate a term in a sequence from one or more previous terms in the sequence. Formulas that can be used to do this are called recursion formulas. A recursion formula consists of at least two parts. The parts give the value(s) of the first term(s) in the sequence, and an equation that can be used to calculate each of the other terms from the term(s) before it. An example is the formula t1 = 5 tn = tn − 1 + 2 The first part of the formula shows that the first term is 5. The second part of the formula shows that each term after the first term is found by adding 2 to the previous term. The sequence is 5, 7, 9, 11, 13, … Thus, this recursion formula determines the same sequence as the explicit formula tn = 2n + 3.
6.4 Recursion Formulas • MHR 457
I NVESTIGATE & I NQUIRE In one of his books, the great Italian mathematician Leonardo Fibonnaci (c.1180−c.1250) described the following situation. A pair of rabbits one month old is too young to produce more rabbits. But at the end of the second month, they produce a pair of rabbits, and a pair of rabbits every month after that. Each new pair of rabbits does the same thing, producing a pair of rabbits every month, starting at the end of the second month. The table shows how a family of rabbits grows. Copy and extend the table to find the number of pairs of rabbits at the start of the ninth month. 1.
Start of Month
Number of Pairs
1
1
2
1
3
2
4
3
5
5
Diagram
List the numbers of pairs of rabbits in order as the first 9 terms of a sequence. This sequence is known as the Fibonnaci sequence. After the first two terms of the sequence, how can each term be calculated from previous terms? 2.
3. Use the following recursion formula to write the first 9 terms of a sequence. Compare the result with the Fibonnaci sequence. t1 = 1 t2 = 1 tn = tn − 1 + tn − 2 4.
Is the Fibonnaci sequence arithmetic, geometric, or neither? Explain.
5.
Is the Fibonnaci sequence a function? Explain.
458 MHR • Chapter 6
EXAMPLE 1 Writing an Arithmetic Sequence From a Recursion Formula Write the first 5 terms of the sequence determined by the recursion formula. t1 = 11 tn = tn − 1 − 4 SOLUTION From the first part of the recursion formula, the first term is 11. From the second part of the recursion formula, After finding the second term, as shown on the screen, press the ENTER key repeatedly to find more terms. t2 = t1 − 4 t3 = t2 − 4 = 11 − 4 =7−4 =7 =3 t4 = t3 − 4 t5 = t4 − 4 =3−4 = −1 − 4 = −1 = −5 The first 5 terms of the sequence are 11, 7, 3, −1, and −5. Note that the sequence in Example 1 is an arithmetic sequence with a first term of 11 and a common difference of −4. The sequence could be determined by the explicit formula tn = 15 − 4n or f(n) = 15 − 4n. EXAMPLE 2 Writing a Geometric Sequence From a Recursion Formula Write the first 5 terms of the sequence determined by the recursion formula. t1 = 2 tn = −3tn − 1 SOLUTION From the first part of the recursion formula, the first term is 2. From the second part of the recursion formula, t2 = −3 × t1 t3 = −3 × t2 = −3 × 2 = −3 × (−6) = −6 = 18 t4 = −3 × t3 t5 = −3 × t4 = −3 × 18 = −3 × (−54) = −54 = 162 The first 5 terms of the sequence are 2, −6, 18, −54, and 162. 6.4 Recursion Formulas • MHR 459
Note that the sequence in Example 2 is a geometric sequence with a first term of 2 and a common ratio of −3. The sequence could be determined by the explicit formula tn = 2(−3)n − 1 or f(n) = 2(−3)n − 1. The graphing calculator screens in the examples show how the ENTER key can be used to write terms of a sequence recursively. However, this method is laborious if you want to write many terms. Another method is to use the mode settings to choose the Seq(sequence) graphing mode and the dot mode, and then to use the sequence Y= editor to enter the sequence as a function. The TABLE key can be used to display the terms of the sequence.
Key
Concepts
• A formula for calculating the nth term of a sequence without knowing any previous terms is called an explicit formula. • A formula used to calculate a term in a sequence from one or more previous terms is called a recursion formula. • A recursion formula consists of at least two parts. The first part(s) give the value(s) of the first term(s) in the sequence. The last part is an equation that can be used to calculate each of the other terms from the term(s) before it. Communicate
Yo u r
Understanding
Describe the difference between a recursion formula and an explicit formula. Describe how finding the 15th term of a sequence using an explicit formula is different from finding the 15th term using a recursion formula. 3. Explain why a recursion formula must have at least two parts. 4. Describe how you would write an explicit formula for the sequence determined by the recursion formula t1 = 2; tn = tn − 1 + 4. 5. Explain why the recursion formula t1 = 1; tn = 2tn − 1 determines the same sequence as the recursion formula f(1) = 1; f(n) = 2f (n − 1). 1. 2.
460 MHR • Chapter 6
Practise A 1. Use the recursion formula to write the first 5 terms of each sequence. a) t1 = 4; tn = tn − 1 + 3 b) t1 = 3; tn = tn − 1 − 2 c) t1 = −1; tn = 2tn − 1 d) t1 = 48; tn = 0.5tn − 1 e) t1 = 6; tn = tn − 1 + 2n 2 f) t1 = −2; tn = 4tn − 1 + n g) t1 = 2; tn = tn − 1 − 2n + 1 h) t1 = −3; tn = 4 − 2tn − 1 2. Use the recursion formula to write the first 5 terms of each sequence. a) t1 = 3; t2 = 5; tn = tn − 2 − tn − 1 b) t1 = −2; t2 = 3; tn = 2tn − 2 + tn − 1 c) t1 = 2; t2 = 1; tn = tn − 2 + tn − 1 − 2n d) t1 = 1; t2 = −2; tn = tn − 2 × tn − 1 e) t1 = −1; t2 = 2; t3 = 4; tn = 2tn − 3 − tn − 2 + 3tn − 1 2 2 f) t1 = 1; t2 = 1; tn = (tn − 2) + (tn − 1)
Use the recursion formula to write the first 6 terms of each sequence. a) f(1) = 12; f(n) = f(n − 1) + 6 b) f(1) = 4; f(n) = 3f (n − 1) c) f(1) = 1.5; f(2) = 2.5; f(n) = f(n − 1) + f(n − 2) d) f(1) = −1; f(2) = 1; f(n) = f(n − 1) ÷ f(n − 2) 3.
Use the recursion formula to write the first 6 terms of each sequence. a) t1 = 5; tn + 1 = tn + 1 tn b) t1 = 80; tn + 1 = 2 c) t1 = −1; tn + 1 = 3(tn + 1) 4.
d)
t1 = 1; t2 = 1; tn + 2 = tn + 1 − tn
Write an explicit formula for the sequence determined by each recursion formula. a) t1 = 1; tn = tn − 1 − 4 b) t1 = 2; tn = 3tn − 1 1 c) t1 = 20; tn = – tn – 1 2 1 1 d) t1 = ; tn = tn – 1 2 2 5.
Apply, Solve, Communicate Verify that the sequence determined by the recursion formula t1 = 10; tn = tn − 1 − 2 is arithmetic. 6.
Verify that the sequence determined by the recursion formula t1 = 20; tn = 0.5tn − 1 is geometric. 7.
Is the sequence determined by the recursion formula t1 = 5; tn = tn − 1 + n − 4 arithmetic, geometric, or neither? Explain. 8.
6.4 Recursion Formulas • MHR 461
B 9. Shading pattern
The first 4 diagrams
in a pattern are shown. a) Write the numbers of shaded squares in the first 6 diagrams as a sequence. b) Verify that the recursion formula t1 = 1; tn = tn − 1 + 4n − 3 determines the sequence. c) Use the recursion formula to predict the number of shaded squares in the seventh diagram; the eighth diagram. The numbers of seats in the rows of a theatre are represented by the recursion formula t1 = 20; tn = tn − 1 + 2. a) Determine the number of seats in each of the first 8 rows. b) Write an explicit formula to represent the number of seats in the nth row. 10. Theatre seats
Hans bought a computer for $3000. The value of the computer at the start of each year, beginning from the date of purchase, is determined by the recursion formula t1 = 3000; tn = 0.4tn − 1. a) Determine the value of the computer at the beginning of each year for the first six years. b) What is the annual rate of depreciation of the computer? c) Write an explicit formula for the nth term of the sequence. 11. Application
Write the first 6 terms of the sequence determined by the recursion formula t1 = 1; tn = tn − 1 + 2n − 1. b) What do all the numbers in the sequence have in common? c) Write an explicit formula for the nth term of the sequence. 12. a)
Write an explicit formula for the sequence determined by each of the following recursion formulas. 1 1 2 a) t1 = ; tn = tn – 1 + b) t1 = 0; tn = tn – 1 + 2 n(n +1) n(n +1) 1 c) t1 = 2; tn = tn – 1 − n(n −1) 13.
Use the explicit formula tn = n2 − 5n + 2 and the recursion formula t1 = −2; tn = tn − 1 + 2(n − 3) to find the eighth term of a sequence. a) Which method do you prefer if you use paper and pencil? Explain. b) Which method do you prefer if you use a graphing calculator? Explain. 14. Communication
462 MHR • Chapter 6
C Write the first 5 terms of a sequence determined by the t 1 recursion formula t1 = x and tn = x × n –. 2 b) Write an explicit formula for the nth term of the sequence.
15. Algebra a)
Find the sum of the first 600 terms in the sequence defined by the following recursion formula. t1 = 2 t2 = 5 tn = tn − 1 − tn − 2 16.
Write an explicit formula for the nth term of a geometric sequence determined by each of the following recursion formulas. t1 = 1; tn = 2tn − 1 t1 = −5; tn = 3tn − 1 t1 = 4; tn = 0.5tn − 1 b) Describe the pattern in the relationship between the two formulas for each sequence. c) Use the pattern to predict the explicit formula for the geometric sequence defined by the recursion formula t1 = 1000; tn = 0.1tn − 1. d) Check your prediction from part c). 17. Inquiry/Problem Solving a)
Write an explicit formula for the nth term of an arithmetic sequence determined by each of the following recursion formulas. t1 = 5; tn = tn − 1 + 2 t1 = 6; tn = tn − 1 − 2 t1 = −8; tn = tn − 1 + 5 b) Describe the pattern in the relationship between the two formulas for each sequence. c) Use the pattern to predict the explicit formula for the arithmetic sequence defined by the recursion formula t1 = 19; tn = tn − 1 − 6. d) Check your prediction from part c). 18. a)
6.4 Recursion Formulas • MHR 463
A ball is dropped from an initial height of 2 m. In each bounce, the ball rebounds to 70% of its height at the start of the previous bounce. a) Determine the height of the ball after each of the first five bounces. b) Write a recursion formula that represents the height of the ball after each bounce. c) Write an explicit formula for the sequence. 19. Communication
A
The tower of Hanoi puzzle involves moving a tower of disks from one of three pegs to another in the least number of moves. The disks have different sizes, as shown in the diagram, and must be moved according to a set of rules, as follows. • Transfer only one disk in each move. • Never place a larger disk above a smaller disk on the same peg. 20. Tower of Hanoi
B
C
If there is only one disk on peg A, it can be transferred to peg C in one move. a) If there are two disks on peg A, they can be transferred to peg C in three moves. Describe how. b) Find the least number of moves required to transfer three disks, four disks, and five disks from peg A to peg C. c) Describe how the least number of moves can be calculated using recursion. d) Write an explicit formula for finding the least number of moves for a given number of disks. e) What is the least number of moves for 8 disks?
A C H I E V E M E N T Check
Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application
Maria’s business plan for her new catering company included annual profit projections that could be modelled by the recursion formula P1 = 10 000; Pn = 1.25Pn − 1. a) What was her projected percent increase in profit from one year to the next? b) What was the projected dollar increase in profit from the fourth year to the fifth year? c) Could the formula be valid indefinitely? Explain. d) Write an explicit formula for the nth term of the same annual profit projections.
464 MHR • Chapter 6
6.4 Recursion Formulas • MHR 457
I NVESTIGATE & I NQUIRE In one of his books, the great Italian mathematician Leonardo Fibonnaci (c.1180−c.1250) described the following situation. A pair of rabbits one month old is too young to produce more rabbits. But at the end of the second month, they produce a pair of rabbits, and a pair of rabbits every month after that. Each new pair of rabbits does the same thing, producing a pair of rabbits every month, starting at the end of the second month. The table shows how a family of rabbits grows. Copy and extend the table to find the number of pairs of rabbits at the start of the ninth month. 1.
Start of Month
Number of Pairs
1
1
2
1
3
2
4
3
5
5
Diagram
List the numbers of pairs of rabbits in order as the first 9 terms of a sequence. This sequence is known as the Fibonnaci sequence. After the first two terms of the sequence, how can each term be calculated from previous terms? 2.
3. Use the following recursion formula to write the first 9 terms of a sequence. Compare the result with the Fibonnaci sequence. t1 = 1 t2 = 1 tn = tn − 1 + tn − 2 4.
Is the Fibonnaci sequence arithmetic, geometric, or neither? Explain.
5.
Is the Fibonnaci sequence a function? Explain.
458 MHR • Chapter 6
EXAMPLE 1 Writing an Arithmetic Sequence From a Recursion Formula Write the first 5 terms of the sequence determined by the recursion formula. t1 = 11 tn = tn − 1 − 4 SOLUTION From the first part of the recursion formula, the first term is 11. From the second part of the recursion formula, After finding the second term, as shown on the screen, press the ENTER key repeatedly to find more terms. t2 = t1 − 4 t3 = t2 − 4 = 11 − 4 =7−4 =7 =3 t4 = t3 − 4 t5 = t4 − 4 =3−4 = −1 − 4 = −1 = −5 The first 5 terms of the sequence are 11, 7, 3, −1, and −5. Note that the sequence in Example 1 is an arithmetic sequence with a first term of 11 and a common difference of −4. The sequence could be determined by the explicit formula tn = 15 − 4n or f(n) = 15 − 4n. EXAMPLE 2 Writing a Geometric Sequence From a Recursion Formula Write the first 5 terms of the sequence determined by the recursion formula. t1 = 2 tn = −3tn − 1 SOLUTION From the first part of the recursion formula, the first term is 2. From the second part of the recursion formula, t2 = −3 × t1 t3 = −3 × t2 = −3 × 2 = −3 × (−6) = −6 = 18 t4 = −3 × t3 t5 = −3 × t4 = −3 × 18 = −3 × (−54) = −54 = 162 The first 5 terms of the sequence are 2, −6, 18, −54, and 162. 6.4 Recursion Formulas • MHR 459
Note that the sequence in Example 2 is a geometric sequence with a first term of 2 and a common ratio of −3. The sequence could be determined by the explicit formula tn = 2(−3)n − 1 or f(n) = 2(−3)n − 1. The graphing calculator screens in the examples show how the ENTER key can be used to write terms of a sequence recursively. However, this method is laborious if you want to write many terms. Another method is to use the mode settings to choose the Seq(sequence) graphing mode and the dot mode, and then to use the sequence Y= editor to enter the sequence as a function. The TABLE key can be used to display the terms of the sequence.
Key
Concepts
• A formula for calculating the nth term of a sequence without knowing any previous terms is called an explicit formula. • A formula used to calculate a term in a sequence from one or more previous terms is called a recursion formula. • A recursion formula consists of at least two parts. The first part(s) give the value(s) of the first term(s) in the sequence. The last part is an equation that can be used to calculate each of the other terms from the term(s) before it. Communicate
Yo u r
Understanding
Describe the difference between a recursion formula and an explicit formula. Describe how finding the 15th term of a sequence using an explicit formula is different from finding the 15th term using a recursion formula. 3. Explain why a recursion formula must have at least two parts. 4. Describe how you would write an explicit formula for the sequence determined by the recursion formula t1 = 2; tn = tn − 1 + 4. 5. Explain why the recursion formula t1 = 1; tn = 2tn − 1 determines the same sequence as the recursion formula f(1) = 1; f(n) = 2f (n − 1). 1. 2.
460 MHR • Chapter 6
Practise A 1. Use the recursion formula to write the first 5 terms of each sequence. a) t1 = 4; tn = tn − 1 + 3 b) t1 = 3; tn = tn − 1 − 2 c) t1 = −1; tn = 2tn − 1 d) t1 = 48; tn = 0.5tn − 1 e) t1 = 6; tn = tn − 1 + 2n 2 f) t1 = −2; tn = 4tn − 1 + n g) t1 = 2; tn = tn − 1 − 2n + 1 h) t1 = −3; tn = 4 − 2tn − 1 2. Use the recursion formula to write the first 5 terms of each sequence. a) t1 = 3; t2 = 5; tn = tn − 2 − tn − 1 b) t1 = −2; t2 = 3; tn = 2tn − 2 + tn − 1 c) t1 = 2; t2 = 1; tn = tn − 2 + tn − 1 − 2n d) t1 = 1; t2 = −2; tn = tn − 2 × tn − 1 e) t1 = −1; t2 = 2; t3 = 4; tn = 2tn − 3 − tn − 2 + 3tn − 1 2 2 f) t1 = 1; t2 = 1; tn = (tn − 2) + (tn − 1)
Use the recursion formula to write the first 6 terms of each sequence. a) f(1) = 12; f(n) = f(n − 1) + 6 b) f(1) = 4; f(n) = 3f (n − 1) c) f(1) = 1.5; f(2) = 2.5; f(n) = f(n − 1) + f(n − 2) d) f(1) = −1; f(2) = 1; f(n) = f(n − 1) ÷ f(n − 2) 3.
Use the recursion formula to write the first 6 terms of each sequence. a) t1 = 5; tn + 1 = tn + 1 tn b) t1 = 80; tn + 1 = 2 c) t1 = −1; tn + 1 = 3(tn + 1) 4.
d)
t1 = 1; t2 = 1; tn + 2 = tn + 1 − tn
Write an explicit formula for the sequence determined by each recursion formula. a) t1 = 1; tn = tn − 1 − 4 b) t1 = 2; tn = 3tn − 1 1 c) t1 = 20; tn = – tn – 1 2 1 1 d) t1 = ; tn = tn – 1 2 2 5.
Apply, Solve, Communicate Verify that the sequence determined by the recursion formula t1 = 10; tn = tn − 1 − 2 is arithmetic. 6.
Verify that the sequence determined by the recursion formula t1 = 20; tn = 0.5tn − 1 is geometric. 7.
Is the sequence determined by the recursion formula t1 = 5; tn = tn − 1 + n − 4 arithmetic, geometric, or neither? Explain. 8.
6.4 Recursion Formulas • MHR 461
B 9. Shading pattern
The first 4 diagrams
in a pattern are shown. a) Write the numbers of shaded squares in the first 6 diagrams as a sequence. b) Verify that the recursion formula t1 = 1; tn = tn − 1 + 4n − 3 determines the sequence. c) Use the recursion formula to predict the number of shaded squares in the seventh diagram; the eighth diagram. The numbers of seats in the rows of a theatre are represented by the recursion formula t1 = 20; tn = tn − 1 + 2. a) Determine the number of seats in each of the first 8 rows. b) Write an explicit formula to represent the number of seats in the nth row. 10. Theatre seats
Hans bought a computer for $3000. The value of the computer at the start of each year, beginning from the date of purchase, is determined by the recursion formula t1 = 3000; tn = 0.4tn − 1. a) Determine the value of the computer at the beginning of each year for the first six years. b) What is the annual rate of depreciation of the computer? c) Write an explicit formula for the nth term of the sequence. 11. Application
Write the first 6 terms of the sequence determined by the recursion formula t1 = 1; tn = tn − 1 + 2n − 1. b) What do all the numbers in the sequence have in common? c) Write an explicit formula for the nth term of the sequence. 12. a)
Write an explicit formula for the sequence determined by each of the following recursion formulas. 1 1 2 a) t1 = ; tn = tn – 1 + b) t1 = 0; tn = tn – 1 + 2 n(n +1) n(n +1) 1 c) t1 = 2; tn = tn – 1 − n(n −1) 13.
Use the explicit formula tn = n2 − 5n + 2 and the recursion formula t1 = −2; tn = tn − 1 + 2(n − 3) to find the eighth term of a sequence. a) Which method do you prefer if you use paper and pencil? Explain. b) Which method do you prefer if you use a graphing calculator? Explain. 14. Communication
462 MHR • Chapter 6
C Write the first 5 terms of a sequence determined by the t 1 recursion formula t1 = x and tn = x × n –. 2 b) Write an explicit formula for the nth term of the sequence.
15. Algebra a)
Find the sum of the first 600 terms in the sequence defined by the following recursion formula. t1 = 2 t2 = 5 tn = tn − 1 − tn − 2 16.
Write an explicit formula for the nth term of a geometric sequence determined by each of the following recursion formulas. t1 = 1; tn = 2tn − 1 t1 = −5; tn = 3tn − 1 t1 = 4; tn = 0.5tn − 1 b) Describe the pattern in the relationship between the two formulas for each sequence. c) Use the pattern to predict the explicit formula for the geometric sequence defined by the recursion formula t1 = 1000; tn = 0.1tn − 1. d) Check your prediction from part c). 17. Inquiry/Problem Solving a)
Write an explicit formula for the nth term of an arithmetic sequence determined by each of the following recursion formulas. t1 = 5; tn = tn − 1 + 2 t1 = 6; tn = tn − 1 − 2 t1 = −8; tn = tn − 1 + 5 b) Describe the pattern in the relationship between the two formulas for each sequence. c) Use the pattern to predict the explicit formula for the arithmetic sequence defined by the recursion formula t1 = 19; tn = tn − 1 − 6. d) Check your prediction from part c). 18. a)
6.4 Recursion Formulas • MHR 463
A ball is dropped from an initial height of 2 m. In each bounce, the ball rebounds to 70% of its height at the start of the previous bounce. a) Determine the height of the ball after each of the first five bounces. b) Write a recursion formula that represents the height of the ball after each bounce. c) Write an explicit formula for the sequence. 19. Communication
A
The tower of Hanoi puzzle involves moving a tower of disks from one of three pegs to another in the least number of moves. The disks have different sizes, as shown in the diagram, and must be moved according to a set of rules, as follows. • Transfer only one disk in each move. • Never place a larger disk above a smaller disk on the same peg. 20. Tower of Hanoi
B
C
If there is only one disk on peg A, it can be transferred to peg C in one move. a) If there are two disks on peg A, they can be transferred to peg C in three moves. Describe how. b) Find the least number of moves required to transfer three disks, four disks, and five disks from peg A to peg C. c) Describe how the least number of moves can be calculated using recursion. d) Write an explicit formula for finding the least number of moves for a given number of disks. e) What is the least number of moves for 8 disks?
A C H I E V E M E N T Check
Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application
Maria’s business plan for her new catering company included annual profit projections that could be modelled by the recursion formula P1 = 10 000; Pn = 1.25Pn − 1. a) What was her projected percent increase in profit from one year to the next? b) What was the projected dollar increase in profit from the fourth year to the fifth year? c) Could the formula be valid indefinitely? Explain. d) Write an explicit formula for the nth term of the same annual profit projections.
464 MHR • Chapter 6
