Understanding Arithmetic and Geometric Sequences: Rules and Conversions

Arithmetic Sequences

Arithmetic sequences are number patterns where the difference between consecutive terms is constant. You'll need to master two key ways to describe these patterns:

1. Explicit rules let you find any term directly using its position number
2. Recursive rules define each term based on the previous term

Learning these rules will help you solve problems more efficiently and recognize patterns in math and real-world situations.

Arithmetic vs. Geometric Sequences

Arithmetic and geometric sequences follow different patterns that are easy to spot once you know what to look for.

In an arithmetic sequence, you add or subtract the same value each time. For example, 2, 5, 8, 11, 14... increases by 3 each time, while 15, 10, 5, 0... decreases by 5 each time.

In a geometric sequence, you multiply or divide by the same value each time. For example, 1, 3, 9, 27... multiplies by 3 each time, while 4, 2, 1, 1/2... divides by 2 each time.

Quick Tip: To check if a sequence is arithmetic, find the difference between consecutive terms - if it's always the same, you've got an arithmetic sequence!

Recursive and Explicit Rules

Recursive and explicit rules give you different ways to find terms in an arithmetic sequence.

The recursive rule describes each term based on the previous one: $a_n = a_{n-1} + d$, where $a_1$ is the first term and $d$ is the common difference. For example, in 62, 69, 76, 83, 90..., the rule is $a_n = a_{n-1} + 7$, where $a_1 = 62$.

The explicit rule lets you find any term directly: $a_n = a_1 + (n-1)d$. For the sequence 83, 61, 39, 17..., the rule is $a_n = 105 - 22n$ (simplified from $a_n = 83 + (n-1)(-22)$).

Remember: The explicit rule is particularly useful when you need to find terms far along in the sequence without calculating all the terms in between!

Converting Between Rules

Converting between recursive and explicit rules is straightforward once you identify the first term ($a_1$) and common difference ($d$).

To convert an explicit rule like $a_n = -6 + (n-1)(12)$ to recursive form:

• Identify $a_1 = -6$ and $d = 12$
• Write as $a_n = a_{n-1} + 12$, where $a_1 = -6$

To convert a recursive rule like $a_n = a_{n-1} - 4$, where $a_1 = 29$ to explicit form:

• Use $a_n = a_1 + (n-1)d$ with $a_1 = 29$ and $d = -4$
• This gives $a_n = 29 + (n-1)(-4)$

Pro Tip: If you have an explicit rule in the form $a_n = b + cn$, you'll need to rearrange it to find $a_1$. Just plug in n=1 to find your first term!

Practice Examples

Here are some examples of finding rules for different arithmetic sequences:

For the sequence -22.7, -18.4, -14.1, -9.8, -5.6:

• The common difference is +4.3
• Recursive rule: $a_n = a_{n-1} + 4.3$, where $a_1 = -22.7$

For 11, 8.5, 6, 3.5, 1:

• Common difference is -2.5
• Explicit rule: $a_n = 13.5 - 2.5n$

For 5, 11, 17, 23, 29:

• Common difference is +6
• This gives $a_n = -1 + 6n$

You can always check your work by calculating a few terms and making sure they match the original sequence.

Confidence Booster: If you can identify the pattern and the first term, you've already done most of the work! The rules are just formal ways to express what you already see.