Change in Arithmetic | undefined Unit 2 Study Guide

The Core Idea: Change in Arithmetic and Geometric Sequences

In mathematics, a sequence is an ordered list of numbers. This topic explores two specific types of sequences that are fundamental to understanding patterns of change: arithmetic and geometric sequences. The core idea is to identify, describe, and predict the behavior of sequences that follow a consistent, repeating rule of change.

An arithmetic sequence is characterized by a constant additive change. This means that to get from one term to the next, you always add the same fixed number, known as the common difference. This represents a linear pattern of growth or decay. In contrast, a geometric sequence is characterized by a constant multiplicative change. To get from one term to the next, you always multiply by the same fixed number, known as the common ratio. This represents an exponential pattern of growth or decay. By understanding these two simple rules of progression, we can develop powerful formulas to find the value of any term in the sequence without having to list all the terms that come before it.

Key Formulas

The behavior of arithmetic and geometric sequences can be described by explicit formulas that allow for the direct calculation of any term, denoted as the $n$ -th term. These formulas depend on the first term of the sequence and the constant pattern of change (the common difference or common ratio).

Arithmetic Sequence Formula

The $n$ -th term of an arithmetic sequence, denoted $a_{n}$ , can be found using the following formula:

$a_{n} = a_{1} + (n - 1) d$

Where:

$a_{n}$ is the term you want to find (the $n$ -th term).
$a_{1}$ is the first term of the sequence.
$n$ is the term number or position in the sequence (e.g., 5th, 20th, 100th).
$d$ is the common difference, the constant value added to get from one term to the next. It is calculated as $d = a_{k + 1} - a_{k}$ for any two consecutive terms.

Geometric Sequence Formula

The $n$ -th term of a geometric sequence, denoted $g_{n}$ , can be found using the following formula:

$g_{n} = g_{1} \cdot r^{n - 1}$

Where:

$g_{n}$ is the term you want to find (the $n$ -th term).
$g_{1}$ is the first term of the sequence.
$n$ is the term number or position in the sequence.
$r$ is the common ratio, the constant value multiplied to get from one term to the next. It is calculated as $r = \frac{g _{k + 1}}{g _{k}}$ for any two consecutive terms.

Understanding Indexing

A critical detail in working with sequences is the concept of indexing. The index refers to the set of integers used to number the terms. The standard formulas presented above assume the sequence begins with an index of $n = 1$ , meaning the first term is $a_{1}$ or $g_{1}$ . However, it is possible for a sequence to begin with an index of $n = 0$ or any other integer.

The choice of the starting index directly affects the formula for the $n$ -th term. If the first term of a sequence is indexed by $n = 0$ , the formulas must be adjusted accordingly.

Arithmetic Sequence starting at $n = 0$ :
If the first term is $a_{0}$ , the formula for the $n$ -th term becomes:
$a_{n} = a_{0} + n d$
Notice that the $(n - 1)$ term is replaced by $n$ . This is because to get to the $n$ -th term from the 0-th term, you must add the common difference $d$ a total of $n$ times.
Geometric Sequence starting at $n = 0$ :
If the first term is $g_{0}$ , the formula for the $n$ -th term becomes:
$g_{n} = g_{0} \cdot r^{n}$
Similarly, the exponent $(n - 1)$ is replaced by $n$ . To get to the $n$ -th term from the 0-th term, you must multiply by the common ratio $r$ a total of $n$ times.

Always check the problem statement to determine the index of the first given term. If a problem provides a sequence like $a_{1}, a_{2}, a_{3}, ...$ , use the standard formulas. If it specifies that the first term corresponds to $n = 0$ , you must use the adjusted formulas.

Core Concepts & Rules

Arithmetic Sequence: A sequence where the difference between any two consecutive terms is a constant. This constant is called the common difference ( $d$ ).
Geometric Sequence: A sequence where the ratio of any two consecutive terms is a constant. This constant is called the common ratio ( $r$ ).
Identifying an Arithmetic Sequence: To determine if a sequence is arithmetic, subtract each term from the term that follows it. If the result is always the same, the sequence is arithmetic.
- Example: In the sequence $5, 2, - 1, - 4, ...$ , the difference is consistently $- 3$ .
Identifying a Geometric Sequence: To determine if a sequence is geometric, divide each term by the term that precedes it. If the result is always the same, the sequence is geometric.
- Example: In the sequence $100, 50, 25, 12.5, ...$ , the ratio is consistently $0.5$ .
The $n$ -th Term Formula (Arithmetic): The formula $a_{n} = a_{1} + (n - 1) d$ allows for the direct calculation of any term in an arithmetic sequence, given the first term ( $a_{1}$ ) and the common difference ( $d$ ).
The $n$ -th Term Formula (Geometric): The formula $g_{n} = g_{1} \cdot r^{n - 1}$ allows for the direct calculation of any term in a geometric sequence, given the first term ( $g_{1}$ ) and the common ratio ( $r$ ).
The Importance of Indexing: The formulas for the $n$ -th term depend on the index of the first term. Be mindful of whether the sequence starts at $n = 1$ , $n = 0$ , or another integer, and adjust the formula as needed.

Step-by-Step Example 1: Finding the Rule for an Arithmetic Sequence

Problem: Consider the sequence $8, 13, 18, 23, ...$

(a) Identify the sequence as arithmetic or geometric and determine its common difference or ratio.

(b) Write an explicit formula for the $n$ -th term, $a_{n}$ .

Step 1: Identify the type of sequence and the common value.

First, test for a common difference (arithmetic):

$13 - 8 = 5$
$18 - 13 = 5$
$23 - 18 = 5$

The difference between consecutive terms is a constant value of 5. Therefore, the sequence is arithmetic.

The common difference, $d$ , is 5.

Step 2: Identify the first term.

The first term given in the sequence is 8. Assuming the sequence starts with an index of $n = 1$ , we have $a_{1} = 8$ .

Step 3: Write the explicit formula for the $n$ -th term.

Use the standard formula for an arithmetic sequence: $a_{n} = a_{1} + (n - 1) d$ .

Substitute the values we found for $a_{1}$ and $d$ :

$a_{n} = 8 + (n - 1) 5$

This is the explicit formula for the $n$ -th term of the sequence.

Step 4: Find the 41st term.

To find the 41st term, we need to calculate $a_{41}$ . We substitute $n = 41$ into our formula:

$a_{41} = 8 + (41 - 1) 5$

$a_{41} = 8 + (40) 5$

$a_{41} = 8 + 200$

$a_{41} = 208$

The 41st term of the sequence is 208.

Step-by-Step Example 2: Exam-Style Application from a Table

Problem: A biologist is tracking the population of a certain bacteria in a lab culture. The population is recorded at the end of each hour, as shown in the table below, where $n = 1$ represents the end of the first hour.

Hour ( $n$ )	Population ( $P_{n}$ )
1	1200
2	1800
3	2700

(a) Determine if the population model is arithmetic or geometric.

(b) Write an explicit formula for the population, $P_{n}$ , at the end of the $n$ -th hour.

Step 1: Test for an arithmetic pattern.

Calculate the difference in population between consecutive hours:

Hour 2 to Hour 1: $1800 - 1200 = 600$
Hour 3 to Hour 2: $2700 - 1800 = 900$

The differences are not constant ( $600 \neq = 900$ ). Therefore, the sequence is not arithmetic.

Step 2: Test for a geometric pattern.

Calculate the ratio of the population between consecutive hours:

Hour 2 to Hour 1: $\frac{1800}{1200} = 1.5$
Hour 3 to Hour 2: $\frac{2700}{1800} = 1.5$

The ratio is constant. Therefore, the sequence is geometric. The common ratio, $r$ , is 1.5.

Step 3: Identify the first term and write the explicit formula.

From the table, the population at the end of the first hour ( $n = 1$ ) is 1200. So, our first term is $g_{1} = 1200$ .

Use the standard formula for a geometric sequence: $g_{n} = g_{1} \cdot r^{n - 1}$ .

Substitute the values for $g_{1}$ and $r$ :

$P_{n} = 1200 \cdot (1.5)^{n - 1}$

This is the explicit formula for the bacteria population.

Step 4: Predict the population at the end of the 8th hour.

To find the population at the end of the 8th hour, we need to calculate $P_{8}$ . Substitute $n = 8$ into the formula:

$P_{8} = 1200 \cdot (1.5)^{8 - 1}$

$P_{8} = 1200 \cdot (1.5)^{7}$

Now, use a calculator to evaluate the expression:

$P_{8} = 1200 \cdot (17.0859375)$

$P_{8} = 20503.125$

Since population must be an integer, we can state the population is approximately 20,503.

Using Your Calculator

The formulas for arithmetic and geometric sequences are primarily analytical. You should be able to set up the correct formula by hand. A calculator is then used as a tool for efficient and accurate computation, especially for large term numbers or complex ratios.

Primary Use: Calculation

The most common use of a calculator for this topic is to evaluate the final expression for an $n$ -th term, particularly in geometric sequences.

Example: Find the 25th term of a geometric sequence with $g_{1} = 10$ and $r = 1.04$ .

Set up the formula:
$g_{25} = 10 \cdot (1.04)^{25 - 1} = 10 \cdot (1.04)^{24}$
Enter into the calculator (TI-84 style):
Type 10 * (1.04) ^ 24 and press ENTER.
Result:
The calculator will display approximately $25.633$ .

Secondary Use: Verification

You can use your calculator to quickly verify if a list of numbers from a table forms an arithmetic or geometric sequence.

Example: Given the terms 4, 10, 25, 62.5.

Test for common ratio:
- Calculate $10/4$ . The result is $2.5$ .
- Calculate $25/10$ . The result is $2.5$ .
- Calculate $62.5/25$ . The result is $2.5$ .
Since the results are the same, it is a geometric sequence.

AP Exam Quick Hit

Common Question Types

Identify and Define: You will be given the first few terms of a sequence, either as a list or in a table. You must first determine if it is arithmetic or geometric by checking for a common difference or ratio, and then write the explicit formula for the $n$ -th term.
- Example: The first four terms of a sequence are 6, 2, -2, -6. Write a rule for the $n$ -th term, $a_{n}$ .
Find a Term from Limited Information: You will be given two non-consecutive terms and asked to find another term or the explicit formula. This requires using the $n$ -th term formula to solve for the common difference/ratio and the first term.
- Example: The 5th term of an arithmetic sequence is 17 and the 11th term is 35. What is the value of the first term, $a_{1}$ ?
Contextual Application: A real-world scenario involving linear or exponential growth/decay will be described. You must model the situation with the correct type of sequence and use the formula to make a prediction.
- Example: The value of a car depreciates by 15% each year. If the car was purchased for $25,000, write a formula for its value after $n$ years and find its value after 6 years. (Note: This may involve starting at index $n = 0$ ).

Common Mistakes

Confusing Arithmetic and Geometric Rules: Applying the common difference formula ( $a_{n} = a_{1} + (n - 1) d$ ) to a geometric sequence, or the common ratio formula ( $g_{n} = g_{1} \cdot r^{n - 1}$ ) to an arithmetic sequence. Always verify the pattern first.
The $(n - 1)$ vs. $n$ Indexing Error: Using $n$ in the exponent or as the multiplier for $d$ when the sequence starts at $n = 1$ , or using $(n - 1)$ when the sequence starts at $n = 0$ . This is a very common error. Always confirm the starting index. For example, writing $a_{n} = a_{1} + n d$ is incorrect if the first term is $a_{1}$ .
Incorrect Calculation of Common Ratio ( $r$ ): Calculating the ratio as $\frac{previous term}{current term}$ instead of the correct $\frac{current term}{previous term}$ . For the sequence $8, 4, 2, ...$ , the ratio is $r = 4/8 = 0.5$ , not $r = 8/4 = 2$ .
Order of Operations Error: In the geometric formula $g_{n} = g_{1} \cdot r^{n - 1}$ , incorrectly multiplying $g_{1}$ by $r$ before applying the exponent. The exponent must be calculated first. For example, $3 \cdot 2^{4}$ is $3 \cdot 16 = 48$ , not $(3 \cdot 2)^{4} = 6^{4} = 1296$ .

Change in Arithmetic and Geometric Sequences - AP PreCalculus Study Guide