Recursive Sequences and Rules for Defining Sequences

Recursive Sequences

• $a_n=a_{n-1}+2$, with $a_1=3$
• This says that we can find the term $a_n$ by taking the previous term, $a_{n-1}$, and adding $2$.
• What are the first four terms of this sequence?

  3, 5, 7, 9, ...

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• What kind of sequence is this?

  Arithmetic

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• What is the general term of this sequence in terms of $n$?

• $a_n=4a_{n-1}$, with $a_1=5$
• We can find the term $a_n$ by taking the previous term, $a_{n-1}$, and multiplying it by $4$.
• What are the first four terms of this sequence?

  5, 20, 80, 320, ...

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• What kind of sequence is this?

  Geometric

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• What is the general term of this sequence in terms of $n$?

• $a_n=a_{n-1}+a_{n-2}$, with $a_1=1,\ a_2 = 1$
• This is called the Fibonacci sequence
• We can find the term $a_n$ by taking the previous term, $a_{n-1}$, and adding the term that came before that, $a_{n-2}$.
• What are the first six terms of this sequence?

  1, 1, 2, 3, 5, 8, ...

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• What kind of sequence is this?

  Neither arithmetic nor geometric

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Example: Creating Rules to Define Sequences

• The numerators are $7, 11, 15, 19, \ \dots$ which form an arithmetic sequence with first term 7 and common difference of 4.
• The general term for the numerator is $a_n = 7 + 4(n-1)$
• The denominators are $4,9,16,25,\ \dots$ which is a sequence of perfect squares $\rightarrow \ \ 2^2, 3^2, 4^2, 5^2,\ \dots$.
• The general term for the denominator is $b_n = (n+1)^2$
• Putting them together, the general term of this sequence is: $t_n = \dfrac{a_n}{b_n} = \dfrac{7 + 4(n-1)}{(n+1)^2}$

Finding First and Second Differences

• The next second difference will be $\bct{+5}$, so the next first difference will be $\bco{13} \bct{+5} = \bco{+18}$.
• Applying this first difference to the last known number of the sequence, we get $\bm{33} \bco{+18} = \bm{\boxed{51}}$.

Recursive Sequences

• $-1-2=-3 \quad \colorThree{\checkmark}$
• $-3-(-1)=-2 \quad \colorThree{\checkmark}$