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# Recursive Formulas

## Where a term is based on prior term(s) in a sequence.

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Recursive and Explicit Formulas

### Vocabulary

##### Complete the chart.
 Word Definition Sequence ________________________________________________________ ______________ a sequence that has a common difference ______________ a sequence that has a common ratio Explicit formula ________________________________________________________ Recursive formula ________________________________________________________ ______________ a subset of the integers: {1,2,3,4,5....}

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What is the difference between a recursive formula and an explicit formula?

### Recursive Formulas

Using the given recursive formulas, identify the next 5 terms in the sequences that follow:

1. a1=3a2=2\begin{align*}a_1 = 3a_2 = -2\end{align*}  and  an=5an1+an2\begin{align*}a_n = -5a_{n-1} + a_{n-2}\end{align*}
2. a1=3\begin{align*}a_1 = 3\end{align*}  and  an=4an1\begin{align*}a_n = 4a_{n-1}\end{align*}
3. a1=4a2=1\begin{align*}a_1 = -4a_2 = 1\end{align*}  and  an=an1+an2\begin{align*}a_n = -a_{n-1} + a_{n-2}\end{align*}

Given the following sequence of numbers find the recursive formula

1. 1, 5, 9, 13, 17
2. -1, 3, 2, 5, 7, 12, 19
3. -4, 16, -64, 256, -1024

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#### Explicit Formulas

Given a recursive definition of an arithmetic or geometric sequence, you can always find an explicit formula, or an equation to represent the th term of the sequence.

What is the equation to find the explicit formula of an arithmetic sequence? ___________________

Geometric sequence? ___________________

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Name the sequence as arithmetic, geometric, or neither.

1. 21,6,18,3,20,2\begin{align*}-21, -6, 18, -3, 20, -2\end{align*}
2. 0,15,25,35,45,1\begin{align*}0,\frac{-1}{5}, \frac{-2}{5}, \frac{-3}{5}, \frac{-4}{5}, -1\end{align*}
3. 1,3,9,27,81,243\begin{align*}1, 3, 9, 27, 81, 243\end{align*}
4. 2,9,2,1,18,2\begin{align*}2, 9, -2, 1, 18, 2\end{align*}

Write the first 5 terms of the arithmetic sequence(explicit).

1. an=7+3(n1)\begin{align*}a_n = -7 + 3(n-1)\end{align*}
2. an=1135(n1)\begin{align*}a_n = 11-\frac{3}{5}(n-1)\end{align*}
3. an=1+17(n1)\begin{align*}a_n = 1 + \frac{1}{7}(n-1)\end{align*}

Write the formula for the explicit (nth)\begin{align*}(n_{th})\end{align*} term of the arithmetic sequence.

1. 7,133,53,1,113,193\begin{align*}-7, \frac{-13}{3}, \frac{-5}{3}, 1, \frac{11}{3}, \frac{19}{3}\end{align*}
2. 6,4,14,24,34,44\begin{align*}6, -4, -14, -24, -34, -44\end{align*}
3. 9,16,23,30,37,44\begin{align*}9, 16, 23, 30, 37, 44\end{align*}

Convert the explicit and rewrite as recursion:

1. an=9(43)(n1)\begin{align*}a_n = 9(\frac{-4}{3})^{(n-1)}\end{align*}
2. an=6(4)(n1)\begin{align*}a_n = -6(-4)^{(n-1)}\end{align*}
3. an=5(5)(n1)\begin{align*}a_n = -5(5)^{(n-1)}\end{align*}

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