Using Recursive Rules for Sequences So far you
- Slides: 12
Using Recursive Rules for Sequences So far you have worked with explicit rules for the nth term of a sequence, such as a n = 3 n – 2 and a n = 3(2) n. An explicit rule gives a n as a function of the term’s position number n in the sequence. In this lesson you will learn another way to define a sequence — by a recursive rule. A recursive rule gives the beginning term or terms of a sequence and then a recursive equation that tells how a n is related to one or more preceding terms.
Evaluating Recursive Rules Write the first five terms of the sequence. Factorial numbers: a 0 = 1, a n = n • a n – 1 Fibonacci sequence: a 1 = 1, a 2 = 1, a n = a n – 2 + a n – 1 SOLUTION a 0 = 1 a 1= 1 • a 0 = 1 • 1 = 1 a 2= 2 • a 1 = 2 • 1 = 2 a 3= a 1 + a 2= 1 + 1 = 2 a 3= 3 • a 2 = 3 • 2 = 6 a 4= a 2 + a 3= 1 + 2 = 3 a 4 = 4 • a 3 = 4 • 6 = 24 a 5= a 3 + a 4= 2 + 3 = 5
Evaluating Recursive Rules Factorial numbers are denoted by a special symbol, !, called a factorial symbol. The expression n! is read “n factorial” and represents the product of all integers from 1 to n. Here are several factorial values. 0! = 1 (by definition) 3! = 3 • 2 • 1 = 6 1! = 1 2! = 2 • 1 = 2 4! = 4 • 3 • 2 • 1 = 24 5! = 5 • 4 • 3 • 2 • 1 = 120
Evaluating Recursive Rules ACTIVITY Developing Concepts 1 2 Investigating Recursive Rules Find the first five terms of each sequence. a 1 = 3 an = an – 1 + 5 a n = 2 a n – 1 Based on the lists of terms you found in Step 1, what type of sequence is the first recursive rule? the second recursive rule?
Writing a Recursive Rule for an Arithmetic Sequence Write the indicated rule for the arithmetic sequence with a 1 = 4 and d = 3. an explicit rule SOLUTION From a previous lesson you know that an explicit rule for the nth term of the arithmetic sequence is: a n = a 1 + (n – 1) d General explicit rule for a n. = a 41 + (n – 1)d 3 Substitute for a 1 and d. = 1 + 3 n Simplify.
Writing a Recursive Rule for an Arithmetic Sequence Write the indicated rule for the arithmetic sequence with a 1 = 4 and d = 3. a recursive rule SOLUTION To find the recursive equation, use the fact that you can obtain a n by adding the common difference d to the previous term. an = an – 1 + d = a n – 1 + 3 d General recursive rule for a n. Substitute for d. A recursive rule for the sequence is a 1 = 4, a n = a n – 1 + 3.
Writing a Recursive Rule for a Geometric Sequence Write the indicated rule for the geometric sequence with a 1 = 3 and r = 0. 1. an explicit rule SOLUTION From previous lesson you know that an explicit rule for the nth term of the geometric sequence is: an = a 1 r n – 1 n – 1 = a 3(0. 1) 1 r General explicit rule for a n. Substitute for a 1 and r.
Writing a Recursive Rule for a Geometric Sequence Write the indicated rule for the geometric sequence with a 1 = 3 and r = 0. 1. a recursive rule SOLUTION To write a recursive rule, use the fact that you can obtain an by multiplying the previous term by r. an = r • an – 1 = (0. 1) r • an – 1 General recursive rule for a n. Substitute for r. A recursive rule for the sequence is a 1 = 3, a n = (0. 1)a n – 1.
Writing a Recursive Rule Write a recursive rule for the sequence 1, 2, 2, 4, 8, 32, … SOLUTION Beginning with the third term in the sequence, each term is the product of the two previous terms. Therefore, a recursive rule is given by: a 1 = 1, a 2 = 2, a n = a n – 2 • a n – 1
Using Recursive Rules in Real Life Fish A lake initially contains 5200 fish. Each year the population declines 30% due to fishing and other causes, and the lake is restocked with 400 fish. Write a recursive rule for the number a n of fish at the beginning of the nth year. How many fish are in the lake at the beginning of the fifth year? SOLUTION Because the population declines 30% each year, 70% of the fish remain in the lake from one year to the next, and new fish are added. Verbal Model Labels Fish at start of New fish = 0. 7 + of nth year (n – 1)st year added Fish at start of nth year = a n Fish at start of (n – 1)st year = a n – 1 New fish added = 400 Algebraic Model a n = (0. 7)a n – 1 + 400
Using Recursive Rules in Real Life FISH A lake initially contains 5200 fish. Each year the population declines 30% due to fishing and other causes, and the lake is restocked with 400 fish. Write a recursive rule for the number a n of fish at the beginning of the nth year. How many fish are in the lake at the beginning of the fifth year? SOLUTION A recursive rule is: a 1 = 5200, a n = (0. 7)a n – 1 + 400 Find a 5: a 5 = (0. 7) a 2659. 6 a 4 n – 1 + 400 = 2261. 72 ≈ 2262 a 4 = (0. 7) a 3228 3 n – 1 + 400 = 2659. 6 a 3 = (0. 7) 4400 a n 2 – 1 + 400 = 3228 a 2 = (0. 7) 5200 a 1 n – 1 + 400 = 4400 There about 2262 fish in the lake at the beginning of the fifth year.
Using Recursive Rules in Real Life FISH A lake initially contains 5200 fish. Each year the population declines 30% due to fishing and other causes, and the lake is restocked with 400 fish. What happens to the population of fish in the lake over time? SOLUTION You can use a graphing calculator to determine what happens to the lake’s fish population over time. Observe that the numbers approach about 1333 as n gets larger. Over time, the population of fish in the lake stabilizes at about 1333 fish.
- Explicit and recursive formula
- Recurisive formula
- Recursive sequence formula
- Recursive vs explicit
- Geometric and arithmetic sequences formulas
- Recursive arithmetic formula
- Lesson 2 recursive formulas for sequences
- Far far from gusty waves figure of speech
- On sour cream walls donations
- In a kingdom far far away
- Far far away city
- Non recursive algorithm examples
- How to write rules for sequences