# Unit 7: Sequences and Functions

## 7.1 Arithmetic Sequences

• Screen 2 defines arithmetic sequence and common difference, and then screen 3 has some questions: (1) What is the common difference, (2) which is an arithmetic sequence?
• Screen 4 introduces terms. First term, second term, $n$th term, etc.
• Screen 5 introduces $a_n$ notation, starting with the recursive formula $a_n = a_{n-1} + d$, with (screen 6) $a_1$ as the first term.
• Screen 7 has a student write the recursive formula for a given arithmetic sequence.
• Screen 10 introduces the explicit formula, $a_n = a_1 + \left(n-1\right)d$.
• Screen 11 has a student find the $n$th term of a sequence, given the explicit formula.
• Screen 13 introduces some word problems, and 14.
• Screen 15 discusses converting between the two forms of formulas here (recursive and explicit).

Arithmetic Sequences

Consider the sequence of numbers $3, 7, 11, 15, 19, \cdots$.

The first term of that sequence is $a_1 = 3$.

The second term of that sequence is $a_2 = 7$. We also know that $a_3=11$, $a_4=15$, and $a_5=19$.

From each term to the next we are adding $4$. This means that the common difference of this sequence is $d=4$. We can find it by subtracting any two terms: $19-15=4$, $15-11=4$, $11-7=4$, and $7-3=4$.

Consider the sequence of numbers $2, 6, 18, 54, \cdots$.

This sequence does not have a common difference, because $6-2=4$ but $18-6=12$. This is not an arithmetic sequence.

Recursive Form of an Arithmetic Sequence

The recursive form of an arithmetic sequence says where the sequence starts, and what happens from one term to the next.

Consider the sequence of numbers $3, 7, 11, 15, 19, \cdots$.

Where does the sequence start? $a_1=3$.

What happens as we move from one term to the next? We add $d=4$. $a_n=a_{n-1}+4$.

\begin{aligned}a_1&=3\\a_n&=a_{n-1}+4\end{aligned}

Explicit Form of an Arithmetic Sequence

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Converting between Recursive and Explicit Forms of an Arithmetic Sequence

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Converting from Recursive to Explicit Forms of an Arithmetic Sequence

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Converting from Explicit to Recursive Forms of an Arithmetic Sequence

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