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1 RECURRENCE RELATIONS
A sequence is a function from the set {n 0 , n 0 + 1, n 0 + 2, …}, where n 0 is an integer, into the set of real numbers. Frequently, n 0 is 0 or 1. If s is a sequence and n is an element of the domain of s, s(n) is frequently denoted by sn and is called the nth^ term of the sequence s.
EXAMPLE 1. For the sequence a defined by an = n^2 + 4n + 3, find the first five terms.
a 1 = _____, a 2 = _____, a 3 = _____, a 4 = _____, a 5 = _____
A recurrence relation (or difference equation) for a sequence s is a formula that expresses sn in terms of one or more of the previous terms of the sequence for all terms from some index on. A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. The initial condition(s) specify the term(s) that precede the first term where the recurrence relation takes effect.
EXAMPLE 2. Find the first four terms of the recurrence relation with initial condition given below. sn = 2sn− 1 + 1, s 1 = 3
s 1 = ____ s 2 = ____ s 3 = ____ s 4 = ____
EXAMPLE 3. For the sequence a in Example 1,
a. complete the following:
a 1 = ____
a 2 = a 1 + ____
a 3 = a 2 + ____
a 4 = a 3 + ____
a 5 = a 4 + ____
b. find a recurrence relation with initial condition for a.
an = an− 1 + ________, a 1 = ____
EXAMPLE 4. Given the sequence b defined by bn = 0.6⋅ 2 n,
a. find b 0 , b 1 , b 2 , and b 3.
b. Find a recurrence relation with initial condition for sequence b.
c. Sketch the graph of sequence b.
EXAMPLE 6. Let F be the sequence such that Fn is the product of the first n positive integers.
a. Find F 5 and F 6.
b. Find a recurrence relation with initial condition for F.
Note: In the sequence F defined above, Fn is more commonly denoted by n! and is called n factorial. Also we define 0! to be 1.#
EXAMPLE 7. Consider the following recurrence relation with initial condition: gn = −gn− 1 + (−1)n+1, g 1 = 1. Find the value of g1538.
(^) Defining 0! to be 1 makes certain formulas nicer in probability and combinatorics.
EXAMPLE 8. Consider the recurrence relation with initial condition given by hn = hn− 1 + n − 1, h 1 = 0
a. Use Excel to find the value of h 78.
A B
1 n h_n
2 1 0
3 =A2+1 (^) =B2+A3− 1
4 Copy formula above down Copy formula above down
A B
1 n h_n
2 1 0
3 2 1
4 3 3
5 4 6
6 5 10
… … …
79 78 3003
… … …
To answer the question, press TBLSET (2nd Window) again and change TblStart to 93 (see left below). Press TABLE (2nd Graph) to display a table of values starting with the answer###^ (see right below).
As a bonus, we can easily obtain sequential graphs. Press the Window key and enter the values below.
Press the Graph key to view the graph in the selected window (see left below). Press the Trace key and use the left arrow and right arrow keys to visit different points on the graph (see right below).
Changing the window (see below)
(^) The Ask feature can also by used here. See the calculator manual for details.
and regraphing (see left below), press the Trace key and enter 93 for n to answer the original question (see right below).
EXAMPLE 9. The Fibonacci sequence f is defined by the following recurrence relation with initial conditions: fn = fn− 1 + fn− 2 , f 0 = 0, f 1 = 1
a. Find the value of f 7.
HOMEWORK
- Find the first four terms of the sequence a defined by an = n^2 − 2 n.
- For the sequence c defined by cn = n!, find the value of c 5 − c 0.
- Consider the sequence b defined by the following recurrence relation with initial condition: bn = bn− 1 + 4, b 1 = 3
a. Find the value of b 5. b. Find a nonrecursive formula for bn.
- Consider the sequence d defined by the following recurrence relation with initial condition: dn = 3dn− 1 + 5, d 0 = 7
a. Find the value of d 5. b. Use Excel to find the value of d 20.
- The Lucas sequence L is defined by the following recurrence relation with initial conditions: Ln = Ln− 1 + Ln− 2 , L 1 = 1, L 2 = 3
Find the tenth term of the Lucas sequence.
- Consider the sequence e defined by the following recurrence relation with initial condition: en = 2en− 1 + 3en− 2 , e 1 = 4, e 2 = 5
a. Use the sequence feature of the TI-83 calculator to find the value of e 20. b. Use Excel to find the value of e 15.
- Consider the sequence g defined by the following recurrence relation with initial condition:
gn =
n 1
g −
, g 1 = 1
Find the value of g 235.
- Consider the sequence h defined by the following recurrence relation with initial condition: hn = 10hn− 1 + 7, h 1 = 6.
Find the value of h 20.
- Consider the sequence fn = 2n^ − 1. Define sequence f with a recurrence relation with initial condition. Assume that the sequence begins with index 0.
ANSWERS
- a 1 = −1, a 2 = 0, a 3 = 1, a 4 = 0
3a. 19 3b. bn = 4n − 1
4a. 2306 4b. 33124451807
6a. 2615088299 6b. 10761682
- fn = 2⋅fn− 1 + 1, f 0 = 0 (other answers possible)
