Recursive definitions

13.2 Recursive Definitions

A recursive formula always uses the preceding term to define the next term of the sequence. Sequences can have the same formula but because they start with a different number, they are different patterns. 1, 3, 5, 7, . . . and 2, 4, 6, 8, . . . have the same common difference but they are certainly not the same sequence because they have different starting values. Point being, you must include the starting number in any recursive definition.

Recursive Definition

1) An initial condition that tells where the sequence starts.
2) A recursion formula that tells how any term of the sequence relates to the preceding term.

Sample of a recursive defintion: 19, 14, 9, 4, . . . Initial condition: t₁ = 19 (look at the beginning) Recursive formula: t_n = t_n-1 - 5 ( each term is 5 less than the term before!)

Note: Formulas in the previous section where explicit formulas!

Bert says" I can't find Ernie, so let's do some math! Do the following problems. Answers at bottom."

Sample problems:

Find the 3rd, 4th and 5th term of each sequence.

1) t₁ = 5; t_n = t_n-1 + 5

2) t₁ = 3; t_n = 2t_n-1 - 1

3) t₁ = 7; t_n = 3t_n-1 + n

Give a recursive definition for each sequence.

4) 5, 8, 11, 14, . . .

5) 1, 4, 13, 40, 121, . . .

6) 1, 3, 6, 10, 15, 21, . . .

The answers:

1) simply add 5 to the preceding term. That would be:

5, 10,

15, 20, 25

2) Multiply preceding term by 2 and add 1. That gives us:
3, 5, 9, 17, 34

3) Multiply the preceding term by 3 and add the current term number.
7, 23, 72, 220, 665

4) Easy one. look at the pattern. It increases by 3 each time!

Answer:
t₁ = 5
t_n = t_n-1 + 3

5) Think of the pattern as 1, 3 + 1, 12 + 1, 39 + 1, 120 + 1, . . .

Now think of it as 1, ( 3^.1 + 1), (3^.4 + 1), (3^.13 +1), (3^.40 +1)

See the pattern! Multiply the preceding term by 3 and add 1!! Cool!

Answer:
t₁ = 1
t_n = 3t_n-1 + 1

6) Write the pattern like: 1, 1 + 2, 3 + 3, 6 + 4, 10 + 5, 15 + 6,

See it!! Add the current term number to the preceding term!! (good thing I am wearing my contacts!!)

Answer:
t₁ = 1
t_n = t_n-1 + n

That's about it for recursive definitions. ( bad pun, does recursive mean to curse again?)

Let's "sail" right along into the next section!

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