MTH120 College Algebra Chapter 9.2

 9.2 Arithmetic Sequences

Arithmetic sequences are a specific type of sequence in mathematics where the difference between consecutive terms is always the same. This constant difference is called the "common difference," denoted as "d." Arithmetic sequences have several important properties and are widely used in various mathematical and real-world applications. Here's an overview of arithmetic sequences:

Arithmetic Sequence Properties:

1. Common Difference (d): In an arithmetic sequence, the difference between any two consecutive terms is always the same. It is denoted as "d."

2. General Term (a_n): The nth term of an arithmetic sequence is typically denoted as "a_n." The general formula to find the nth term is:

   a_n = a_1 + (n - 1) * d

   • "a_1" is the first term of the sequence.
   • "n" is the position of the term.
   • "d" is the common difference.

3. Sum of n Terms (S_n): The sum of the first "n" terms of an arithmetic sequence can be calculated using the formula:

   S_n = (n/2) * [2a_1 + (n - 1) * d]

Example: Arithmetic Sequence

Let's consider an arithmetic sequence with the following information:

First term (a_1) = 3 Common difference (d) = 2

1. To find the nth term of the sequence, you can use the general term formula:

   a_n = a_1 + (n - 1) * d a_n = 3 + (n - 1) * 2

   • For n = 1: a_1 = 3 + (1 - 1) * 2 = 3
   • For n = 2: a_2 = 3 + (2 - 1) * 2 = 5
   • For n = 3: a_3 = 3 + (3 - 1) * 2 = 7

2. To find the sum of the first "n" terms (S_n), you can use the sum formula:

   S_n = (n/2) * [2a_1 + (n - 1) * d] S_n = (n/2) * [2 * 3 + (n - 1) * 2]

   • For n = 1: S_1 = (1/2) * [2 * 3 + (1 - 1) * 2] = 3
   • For n = 2: S_2 = (2/2) * [2 * 3 + (2 - 1) * 2] = 9
   • For n = 3: S_3 = (3/2) * [2 * 3 + (3 - 1) * 2] = 18

Arithmetic sequences are used in various areas of mathematics and science, including algebra, calculus, physics, and finance. They help describe and model phenomena with constant rates of change.

To find the common difference in an arithmetic sequence, you need to examine the differences between consecutive terms. The common difference is the constant value by which each term increases or decreases compared to the previous term. Here's how you can find the common difference with examples:

Example 1: Finding the Common Difference in an Arithmetic Sequence

Consider the following arithmetic sequence:

7, 10, 13, 16, 19, ...

To find the common difference:

1. Calculate the difference between the second term (10) and the first term (7): Common Difference (d) = 10 - 7 = 3

2. Verify that the same difference applies to the other terms as well:

   • Difference between the third term (13) and the second term (10) is 13 - 10 = 3.
   • Difference between the fourth term (16) and the third term (13) is 16 - 13 = 3.
   • Difference between the fifth term (19) and the fourth term (16) is 19 - 16 = 3.

In this sequence, the common difference is 3.

Example 2: Finding the Common Difference in a Problem

Suppose you have an arithmetic sequence, and the first few terms are given:

a_1 = 5 a_2 = 9 a_3 = 13

To find the common difference:

1. Calculate the difference between the second term (a_2) and the first term (a_1): Common Difference (d) = 9 - 5 = 4

2. Verify that the same difference applies to the other terms as well:

   • Difference between the third term (a_3) and the second term (a_2) is 13 - 9 = 4.
   • You have established that the common difference is 4.

In this sequence, the common difference is 4.

By examining the differences between consecutive terms, you can determine the common difference in an arithmetic sequence. This is a fundamental concept in understanding and working with arithmetic sequences.

To write the terms of an arithmetic sequence, you'll need to know the first term (a_1) and the common difference (d). With this information, you can use the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n - 1) * d

Let's explore a couple of examples to illustrate this process:

Example 1: Arithmetic Sequence with Known a_1 and d

Suppose you have an arithmetic sequence with the first term (a_1) equal to 2 and a common difference (d) of 3. You want to find the first five terms of this sequence.

Using the formula for the nth term:

• For n = 1: a_1 = 2 + (1 - 1) * 3 = 2
• For n = 2: a_2 = 2 + (2 - 1) * 3 = 2 + 3 = 5
• For n = 3: a_3 = 2 + (3 - 1) * 3 = 2 + 6 = 8
• For n = 4: a_4 = 2 + (4 - 1) * 3 = 2 + 9 = 11
• For n = 5: a_5 = 2 + (5 - 1) * 3 = 2 + 12 = 14

So, the first five terms of the sequence are 2, 5, 8, 11, 14.

Example 2: Arithmetic Sequence with Given a_1 and One Known Term

Suppose you know the first term (a_1) is 10, and you have a specific term a_5, which is 34. You want to find the common difference (d) and the first five terms of the sequence.

To find the common difference (d), you can use the formula:

a_5 = a_1 + (5 - 1) * d

Substitute the known values:

34 = 10 + 4d

Now, solve for d:

4d = 34 - 10 4d = 24 d = 24 / 4 d = 6

Now that you know the common difference is 6, you can find the first five terms of the sequence:

• For n = 1: a_1 = 10
• For n = 2: a_2 = 10 + (2 - 1) * 6 = 16
• For n = 3: a_3 = 10 + (3 - 1) * 6 = 22
• For n = 4: a_4 = 10 + (4 - 1) * 6 = 28
• For n = 5: a_5 = 10 + (5 - 1) * 6 = 34 (which we already knew)

So, the first five terms of the sequence are 10, 16, 22, 28, 34.

Arithmetic sequences are useful in various mathematical and real-world applications for modeling linear relationships and uniform changes.

Arithmetic sequences can also be defined using recursive formulas, where each term is calculated based on the previous term and the common difference. The recursive formula for an arithmetic sequence is typically expressed as:

a_(n+1) = a_n + d

Here's how to use a recursive formula to find terms in an arithmetic sequence with some examples:

Example 1: Arithmetic Sequence with Recursive Formula

Suppose you have an arithmetic sequence with a_1 = 5 and a common difference d = 3. Using the recursive formula:

a_(n+1) = a_n + d

1. Start with the first term, a_1 = 5.

2. To find the second term (a_2), use the recursive formula:

   a_(2+1) = a_2 + d a_3 = a_2 + 3

   Now, plug in the known values: a_3 = 5 + 3 = 8

3. Continue this process to find more terms:

   • To find the fourth term, use the recursive formula: a_(4+1) = a_4 + 3 a_5 = a_4 + 3

     Using the previous term: a_5 = 8 + 3 = 11

   • To find the fifth term: a_(5+1) = a_5 + 3 a_6 = a_5 + 3

     a_6 = 11 + 3 = 14

So, you have found the first six terms of this arithmetic sequence: 5, 8, 11, 14, 17, 20.

Example 2: Arithmetic Sequence with Known a_1 and Recursive Formula

Suppose you have a known first term (a_1 = 4) and the following recursive formula:

a_(n+1) = a_n + 2

1. Start with the first term, a_1 = 4.

2. To find the second term (a_2), use the recursive formula:

   a_(2+1) = a_2 + 2 a_3 = a_2 + 2

   Now, plug in the known value: a_3 = 4 + 2 = 6

3. Continue this process to find more terms:

   • To find the fourth term: a_(4+1) = a_4 + 2 a_5 = a_4 + 2

     a_5 = 6 + 2 = 8

   • To find the fifth term: a_(5+1) = a_5 + 2 a_6 = a_5 + 2

     a_6 = 8 + 2 = 10

So, you have found the first six terms of this arithmetic sequence with the known a_1 and the given recursive formula: 4, 6, 8, 10, 12, 14.

Recursive formulas for arithmetic sequences are a convenient way to calculate terms when you know the common difference and the first term.

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same. The explicit formula for an arithmetic sequence allows you to calculate any term in the sequence directly, without having to find the preceding terms. The explicit formula for an arithmetic sequence is:

${�}_{�}={�}_1+(�-1)\cdot �$

where:

• ${�}_{�}$ is the nth term of the sequence.
• ${�}_1$ is the first term of the sequence.
• $�$ is the position of the term.
• $�$ is the common difference between consecutive terms.

Let's work through a couple of examples to illustrate the use of explicit formulas for arithmetic sequences:

Example 1: Using the Explicit Formula for an Arithmetic Sequence

Suppose you have an arithmetic sequence with the following information:

• The first term (${�}_1$) is 6.
• The common difference ($�$) is 4.

You want to find the 10th term of the sequence.

Using the explicit formula:

${�}_{�}={�}_1+(�-1)\cdot �$

Substitute the known values:

${�}_{10}=6+(10-1)\cdot 4$ ${�}_{10}=6+9\cdot 4$ ${�}_{10}=6+36$ ${�}_{10}=42$

So, the 10th term of this arithmetic sequence is 42.

Example 2: Using the Explicit Formula to Find Multiple Terms

Suppose you have an arithmetic sequence with the following information:

• The first term (${�}_1$) is 3.
• The common difference ($�$) is 2.

You want to find the first five terms of the sequence.

Using the explicit formula, you can calculate each term as follows:

• For $�=1$: ${�}_1=3+(1-1)\cdot 2=3$

• For $�=2$: ${�}_2=3+(2-1)\cdot 2=5$

• For $�=3$: ${�}_3=3+(3-1)\cdot 2=7$

• For $�=4$: ${�}_4=3+(4-1)\cdot 2=9$

• For $�=5$: ${�}_5=3+(5-1)\cdot 2=11$

So, the first five terms of this arithmetic sequence are 3, 5, 7, 9, 11.

The explicit formula for an arithmetic sequence allows you to find specific terms directly, which is particularly useful when you need to calculate terms further along in the sequence without finding all the preceding terms.

The explicit formula for an arithmetic sequence is a mathematical expression that allows you to calculate any term in the sequence directly, without having to find the preceding terms. It's a formula that relates the term's position ($�$) to the common difference ($�$) and the first term (${�}_1$) of the sequence. The explicit formula for an arithmetic sequence is:

${�}_{�}={�}_1+(�-1)\cdot �$

Where:

• ${�}_{�}$ represents the nth term of the sequence.
• ${�}_1$ is the first term of the sequence.
• $�$ is the position of the term you want to find.
• $�$ is the common difference between consecutive terms in the sequence.

This formula simplifies the process of finding specific terms in an arithmetic sequence. You only need to know the values of ${�}_1$, $�$, and $�$ to calculate the nth term (${�}_{�}$) directly.

For example, if you have an arithmetic sequence with ${�}_1=5$ and a common difference $�=3$, you can use the explicit formula to find the 10th term (${�}_{10}$):

${�}_{10}=5+(10-1)\cdot 3=5+9\cdot 3=5+27=32$

So, the 10th term of the sequence is 32, and you can use this formula to find any term in the sequence with the same common difference and first term.

You can find the number of terms in a finite arithmetic sequence using the formula for the nth term of an arithmetic sequence and the last term of the sequence. Here's the formula to find the number of terms in a finite arithmetic sequence:

$�=\frac{{�}_{�}-{�}_1}{�}+1$

Where:

• $�$ is the number of terms in the sequence.
• ${�}_{�}$ is the last term of the sequence.
• ${�}_1$ is the first term of the sequence.
• $�$ is the common difference between consecutive terms.

Let's go through an example to illustrate how to use this formula:

Example: Finding the Number of Terms in a Finite Arithmetic Sequence

Suppose you have an arithmetic sequence with the following information:

• The first term (${�}_1$) is 3.
• The common difference ($�$) is 4.
• The last term (${�}_{�}$) is 47.

To find the number of terms in this finite arithmetic sequence, you can use the formula:

$�=\frac{{�}_{�}-{�}_1}{�}+1$

Substitute the known values:

$�=\frac{47-3}{4}+1$

Now, perform the calculations:

$�=\frac{44}{4}+1$

$�=11+1$

$�=12$

So, the number of terms in this finite arithmetic sequence is 12.

This formula is helpful for determining the size of a finite arithmetic sequence when you know the first term, the common difference, and the last term. It allows you to find the number of terms directly without having to list all the terms in the sequence.

Arithmetic sequences can be used to model and solve real-world problems where a quantity changes by a constant amount over time. These problems often involve situations where there is a steady increase or decrease in some variable. Let's go through a couple of examples to illustrate how to solve application problems using arithmetic sequences:

Example 1: Finding a Term in a Loan Repayment

Suppose you take out a loan of $5,000, and you agree to repay it in monthly installments over 3 years (36 months) with an interest rate that adds $20 to the repayment amount each month.

The amount you repay each month forms an arithmetic sequence. Let's use an arithmetic sequence to find out how much you will repay in the 12th month.

• First term (${�}_1$): Initial loan amount = $5,000
• Common difference ($�$): Increase each month = $20
• Number of months ($�$): 12

The nth term formula for an arithmetic sequence is ${�}_{�}={�}_1+(�-1)�$.

${�}_{12}=5000+(12-1)\times 20$

${�}_{12}=5000+11\times 20$

${�}_{12}=5000+220$

${�}_{12}=5220$

So, you will repay $5,220 in the 12th month.

Example 2: Finding the Sum of a Series - Savings Account

Consider a savings account where you deposit $100 at the beginning of each month, and the interest adds $5 each month. You want to know how much money you will have saved after 24 months.

• First term (${�}_1$): Initial deposit = $100
• Common difference ($�$): Monthly deposit = $5
• Number of months ($�$): 24

The sum formula for an arithmetic series is ${�}_{�}=\frac{�}{2}[2{�}_1+(�-1)�]$.

${�}_{24}=\frac{24}{2}[2\times 100+(24-1)\times 5]$

${�}_{24}=12[200+23\times 5]$

${�}_{24}=12[200+115]$

${�}_{24}=12\times 315$

${�}_{24}=3780$

So, after 24 months, you will have saved $3,780.

These examples demonstrate how arithmetic sequences and series can be applied to model and solve financial problems. In these cases, the key is to identify the first term, common difference, and the number of terms in the sequence or series.

Arithmetic sequences are commonly used to solve various real-world problems where there is a consistent change in a quantity over time. These problems often involve scenarios such as financial planning, distance, or time. Let's explore a couple of examples of solving application problems using arithmetic sequences:

Example 1: Calculating Savings Over Time

Suppose you want to save money over a period of 24 months. You plan to deposit $100 into your savings account at the beginning of each month, and your account earns 2% interest per month. You want to find out how much money you will have saved after 24 months.

To solve this problem, we can use an arithmetic sequence for the monthly savings:

• First term (${�}_1$): Initial deposit = $100
• Common difference ($�$): Monthly deposit = $100
• Number of months ($�$): 24

Now, we'll use the formula for the sum of an arithmetic series:

${�}_{�}=\frac{�}{2}[2{�}_1+(�-1)�]$

Plug in the values:

${�}_{24}=\frac{24}{2}[2\cdot 100+(24-1)\cdot 100]$

${�}_{24}=12[200+23\cdot 100]$

${�}_{24}=12[200+2300]$

${�}_{24}=12\cdot 2500$

${�}_{24}=30,000$

So, after 24 months, you will have saved $30,000.

Example 2: Finding the Distance Traveled

Suppose you are on a road trip and you drive at a constant speed of 60 miles per hour (mph). You want to find out how far you will have traveled after 4 hours.

In this scenario, the distance traveled forms an arithmetic sequence:

• First term (${�}_1$): Initial distance = 0 miles
• Common difference ($�$): Distance covered each hour = 60 miles
• Number of hours ($�$): 4

We can use the arithmetic sequence formula:

${�}_{�}={�}_1+(�-1)�$

Plug in the values:

${�}_4=0+(4-1)\cdot 60$

${�}_4=0+3\cdot 60$

${�}_4=0+180$

${�}_4=180$

So, after 4 hours of driving at a constant speed of 60 mph, you will have traveled 180 miles.

These examples illustrate how arithmetic sequences can be applied to solve problems involving savings and distance over time. In each case, understanding the first term, common difference, and the number of terms is essential to apply the arithmetic sequence formula effectively.

For the following exercises, find the common difference for the arithmetic sequence provided. 1. {5,14,12,23,27,...} 2. {0,12,1,42,2,...} For the following exercises, determine whether the sequence is arithmetic. If so find the common difference. 3. {11.4,4.3,8.5,5.1,3,...} 4. {4,16,62,226,1021,...} For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference. 5. a1=−25 , d=−9 6. a1=0 , d=23 For the following exercises, write the first five terms of the arithmetic series given two terms. 7. a1=14,a7=−32 8. a13=−50,a23=−150

Let's solve each of these exercises step by step:

For the first set of exercises (Finding the Common Difference):

1. {5, 14, 12, 23, 27, ...}

To find the common difference, we look at the differences between consecutive terms:

• 14 - 5 = 9
• 12 - 14 = -2
• 23 - 12 = 11
• 27 - 23 = 4

The common difference is not the same in this sequence. It varies between 9, -2, 11, and 4. Therefore, this is not an arithmetic sequence.

2. {0, 12, 1, 42, 2, ...}

Again, let's find the differences between consecutive terms:

• 12 - 0 = 12
• 1 - 12 = -11
• 42 - 1 = 41
• 2 - 42 = -40

The common difference is not the same in this sequence either. It varies between 12, -11, 41, and -40. This is not an arithmetic sequence.

For the second set of exercises (Determining if the Sequence is Arithmetic and Finding the Common Difference):

3. {11.4, 4.3, 8.5, 5.1, 3, ...}

Let's find the differences between consecutive terms:

• 4.3 - 11.4 = -7.1
• 8.5 - 4.3 = 4.2
• 5.1 - 8.5 = -3.4
• 3 - 5.1 = -2.1

The common difference is not the same in this sequence. It varies between -7.1, 4.2, -3.4, and -2.1. This is not an arithmetic sequence.

4. {4, 16, 62, 226, 1021, ...}

Let's find the differences between consecutive terms:

• 16 - 4 = 12
• 62 - 16 = 46
• 226 - 62 = 164
• 1021 - 226 = 795

The common difference is not the same in this sequence. It varies between 12, 46, 164, and 795. This is not an arithmetic sequence.

For the third set of exercises (Writing the First Five Terms):

5. a_1 = -25, d = -9

Using the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n - 1) * d

Let's find the first five terms:

• For n = 1: a_1 = -25 + (1 - 1) * (-9) = -25
• For n = 2: a_2 = -25 + (2 - 1) * (-9) = -25 - 9 = -34
• For n = 3: a_3 = -25 + (3 - 1) * (-9) = -25 - 18 = -43
• For n = 4: a_4 = -25 + (4 - 1) * (-9) = -25 - 27 = -52
• For n = 5: a_5 = -25 + (5 - 1) * (-9) = -25 - 36 = -61

The first five terms of the sequence are: -25, -34, -43, -52, -61.

6. a_1 = 0, d = 23

Using the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n - 1) * d

Let's find the first five terms:

• For n = 1: a_1 = 0 + (1 - 1) * 23 = 0
• For n = 2: a_2 = 0 + (2 - 1) * 23 = 23
• For n = 3: a_3 = 0 + (3 - 1) * 23 = 46
• For n = 4: a_4 = 0 + (4 - 1) * 23 = 69
• For n = 5: a_5 = 0 + (5 - 1) * 23 = 92

The first five terms of the sequence are: 0, 23, 46, 69, 92.

For the fourth set of exercises (Given Two Terms, Writing the First Five Terms):

7. a_1 = 14, a_7 = -32

Let's use the common difference formula to find the common difference (d):

d = (a_7 - a_1) / (7 - 1) = (-32 - 14) / 6 = (-46) / 6 = -7.67 (approximate)

Since we obtained a non-integer common difference, it is not an arithmetic sequence in the traditional sense. However, we can still write the first five terms:

• For n = 1: a_1 = 14
• For n = 2: a_2 = 14 + 1 * (-7.67) ≈ 6.33
• For n = 3: a_3 = 14 + 2 * (-7.67) ≈ -1.34
• For n = 4: a_4 = 14 + 3 * (-7.67) ≈ -9.67
• For n = 5: a_5 = 14 + 4 * (-7.67) ≈ -17.67

8. a_13 = -50, a_23 = -150

Again, let's use the common difference formula to find the common difference (d):

d = (a_23 - a_13) / (23 - 13) = (-150 - (-50)) / 10 = (-100) / 10 = -10

Now that we have a constant common difference, it is an arithmetic sequence. Let's write the first five terms:

• For n = 1: a_1 = -50
• For n = 2: a_2 = -50 + 1 * (-10) = -60
• For n = 3: a_3 = -50 + 2 * (-10) = -70
• For n = 4: a_4 = -50 + 3 * (-10) = -80
• For n = 5: a_5 = -50 + 4 * (-10) = -90

So, for this arithmetic sequence, the first five terms are: -50, -60, -70, -80, -90.