Arithmetic Sequences: Videos & Practice Problems

Arithmetic sequences are defined by a constant common difference $d$ between consecutive terms. The general term $a_n$ of an arithmetic sequence can be found using the formula $a_n=a_1+\; (n-1)d$, where $a_1$ is the first term. This formula enables efficient calculation of any term, such as the twentieth term, without listing all previous terms. Understanding how to identify the common difference and apply this formula is essential for analyzing sequences and their patterns.

Intro to Arithmetic Sequences

An arithmetic sequence is a type of sequence where each term changes by the same fixed amount, known as the common difference. This common difference, represented by the letter d, is the difference between any two consecutive terms in the sequence. For example, in the sequence 2, 6, 10, 14, the common difference is 4 because each term increases by 4 from the previous term. This means to find the next terms, you simply add 4 repeatedly, resulting in 18 and 22 as the next terms.

To calculate the common difference, subtract the first term from the second term, or generally subtract any term from the term that follows it. Mathematically, this is expressed as:

\[d = a_{n+1} - a_n\]

where \(a_n\) is the nth term and \(a_{n+1}\) is the term immediately after it.

For instance, if the first term \(a_1\) is 10 and the second term \(a_2\) is 8, the common difference is:

\[d = 8 - 10 = -2\]

A negative common difference indicates the sequence is decreasing by that amount each time. To find subsequent terms, subtract the common difference from the previous term. So, the third term \(a_3\) is:

\[a_3 = a_2 + d = 8 + (-2) = 6\]

and the fourth term \(a_4\) is:

\[a_4 = a_3 + d = 6 + (-2) = 4\]

Understanding arithmetic sequences involves recognizing the pattern of constant change and using the common difference to predict future terms. This foundational concept is essential for exploring more complex sequences and series in mathematics.

Find the common difference of the following sequence.
(A) $-9,-4,1,6,\dots$

• A. $4$
• B. $-4$
• C. $-5$
• D. $5$

Answer: (A)

Find the common difference of the following sequence.
(B) $\frac{7}{4},\frac{5}{4},\frac{3}{4},\frac{1}{4},\dots$

• A. $d=-2$
• B. $d=-\frac24$
• C. $2$
• D. $\frac24$

Answer: (B)

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.
(A) $a_1=-12$; $d=-5$

• A. $-12,-17,-22,-27,-32$
• B. $-32,-27,-22,-17,-12$
• C. $-12,-7,-2,3,8$
• D. $8,3,-2,-7,-12$

Answer: (A)

Write the first 5 terms of the arithmetic sequence, given the first term and the common difference.
(B) $a_1=\frac{3}{2}$; $d=\frac{1}{4}$

• A. $\frac32,1,\frac12,0,-\frac12$
• B. $\frac32,\frac54,1,\frac34,\frac12$
• C. $\frac32,\frac74,2,\frac94,\frac52$
• D. $\frac32,2,\frac52,3,\frac72$

Answer: (A)

General Term of an Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted as d. To find the general term, or the nth term, of an arithmetic sequence, we use a specific formula that incorporates the first term and the common difference.

The general term of an arithmetic sequence, represented as a_n, can be calculated using the formula:

\[a_n = a_1 + (n - 1)d\]

Here, a₁ is the first term of the sequence, d is the common difference, and n is the term number. This formula allows you to find any term in the sequence without listing all the previous terms.

For example, consider the arithmetic sequence 2, 6, 10, 14, ... where the first term a₁ is 2 and the common difference d is 4. Plugging these values into the formula gives:

\[a_n = 2 + (n - 1) \times 4\]

This can be simplified by distributing the 4:

\[a_n = 2 + 4n - 4 = 4n - 2\]

Both forms are correct, and the choice between them depends on the context or instructions given.

Using this formula, you can quickly find any term in the sequence. For instance, to find the 20th term (a₂₀), substitute n = 20:

\[a_{20} = 2 + (20 - 1) \times 4 = 2 + 19 \times 4 = 2 + 76 = 78\]

This method is much faster than adding the common difference repeatedly.

Sometimes, you may be given different information, such as the first term and a later term, instead of the common difference. In such cases, you can use the general term formula to find the common difference. For example, if the first term a₁ is 2 and the fifth term a₅ is 14, you can set up the equation:

\[a_5 = a_1 + (5 - 1)d\]

Substituting the known values:

\[14 = 2 + 4d\]

Solving for d:

\[14 - 2 = 4d \implies 12 = 4d \implies d = 3\]

With the common difference found, the general term formula becomes:

\[a_n = 2 + (n - 1) \times 3\]

This approach highlights the importance of understanding the relationship between terms in an arithmetic sequence and how to manipulate the formula to find missing information.

Mastering the general term formula for arithmetic sequences enables efficient calculation of any term and deepens comprehension of sequence patterns, which is fundamental in algebra and various applications in mathematics.

Write a general formula for the arithmetic sequence given.
(A) $-5,-12,-19,-26,\dots$

• A. $a_{n}=7n-12$
• B. $a_{n}=-7n+2$
• C. $a_{n}=7n+2$
• D. $a_{n}=-7n-12$

Answer: (B)

Write a general formula for the arithmetic sequence given.
(B) $\frac{1}{2},\frac{5}{6},\frac{7}{6},\frac{3}{2},\dots$

• A. $a_{n}=\frac13n+\frac56$
• B. $a_{n}=-\frac13n+\frac56$
• C. $a_{n}=\frac13n+\frac26$
• D. $a_{n}=\frac13n+\frac16$

Answer: (D)

Find the indicated term of each arithmetic sequence.
(A) $a_n=7+3(n-1)$
Find $a_{12}$.

• A. $40$
• B. $43$
• C. $37$
• D. $34$

Answer: (A)

Find the indicated term of each arithmetic sequence.
(B) $a_n=\frac{5}{2}+\frac{3(n-1)}{4}$
Find $a_6$.

• A. $\frac{25}{4}$
• B. $5$
• C. $10$
• D. $\frac{25}{2}$

Answer: (A)

General Term of an Arithmetic Sequences Example 1

In an arithmetic sequence, each term is found by adding a constant value called the common difference to the previous term. Given the fifth term as -12 and the fifteenth term as 18, we can determine the common difference and then find the ninth term. The key is to recognize that the difference in term positions corresponds to how many times the common difference is added. Specifically, moving from the fifth term to the fifteenth term involves adding the common difference 10 times, since 15 - 5 = 10.

Using the formula for the nth term of an arithmetic sequence, \(a_n = a_m + (n - m)d\), where \(a_n\) is the nth term, \(a_m\) is the mth term, and \(d\) is the common difference, we substitute the known values:

\$18 = -12 + 10d\(

Solving for \)d\(, add 12 to both sides:

\)18 + 12 = 10d\(

\)30 = 10d\(

Divide both sides by 10:

\)d = 3\(

With the common difference found, we can now calculate the ninth term. Since the ninth term is four positions after the fifth term (9 - 5 = 4), we add the common difference four times to the fifth term:

\)a_9 = a_5 + 4d = -12 + 4(3) = -12 + 12 = 0$

Therefore, the ninth term of the arithmetic sequence is 0. This approach highlights how understanding the relationship between term positions and the common difference allows for efficient calculation of any term in an arithmetic sequence.

General Term of an Arithmetic Sequences Example 2

To determine the number of terms in an arithmetic sequence, consider the sequence 4, 9, 14, 19, ..., with the last term given as 94. Since the number of terms between 19 and 94 is unknown, counting directly is not feasible. Instead, use the general formula for the nth term of an arithmetic sequence:

\[a_n = a_1 + (n - 1)d\]

Here, \(a_n\) represents the nth term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms. For this sequence, the first term \(a_1\) is 4, and the common difference \(d\) is 5, since each term increases by 5 (from 4 to 9, 9 to 14, and so on).

Substituting these values into the formula and setting \(a_n = 94\) gives:

\[94 = 4 + (n - 1) \times 5\]

To solve for \(n\), first subtract 4 from both sides:

\[94 - 4 = (n - 1) \times 5\]

\[90 = 5(n - 1)\]

Next, divide both sides by 5:

\[\frac{90}{5} = n - 1\]

\[18 = n - 1\]

Finally, add 1 to both sides to isolate \(n\):

\[18 + 1 = n\]

\[n = 19\]

Therefore, the arithmetic sequence contains 19 terms. Understanding how to apply the nth term formula allows for efficient calculation of the number of terms in any arithmetic sequence when the first term, common difference, and last term are known.

General Term of an Arithmetic Sequences Example 3

In this problem, ticket sales for a school play form an arithmetic sequence, where the number of tickets sold increases by a constant amount each night. The initial ticket sales on opening night are 120, which represents the first term of the sequence, denoted as a₁. The common difference, d, is 25, indicating that each subsequent night sells 25 more tickets than the previous night.

To find the number of tickets sold on the tenth night, we use the formula for the n-th term of an arithmetic sequence:

\[a_n = a_1 + (n - 1)d\]

Substituting the known values, where a₁ = 120, d = 25, and n = 10, we calculate:

\[a_{10} = 120 + (10 - 1) \times 25 = 120 + 9 \times 25 = 120 + 225 = 345\]

This means that on the tenth night, 345 tickets were sold. Understanding arithmetic sequences and how to apply their formulas allows us to model and predict patterns in real-world scenarios such as ticket sales growth over time.

Here’s what students ask on this topic:

What is an arithmetic sequence and how do you identify it?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted as $d$. To identify an arithmetic sequence, check if the terms increase or decrease by the same amount each time. For example, in the sequence 2, 6, 10, 14, the difference between terms is always 4, so it is an arithmetic sequence with $d=4$. If the difference is negative, like in 10, 8, 6, 4, then the sequence decreases by a constant amount, and $d$ would be negative. Recognizing this pattern is essential for working with arithmetic sequences.

How do you find the common difference in an arithmetic sequence?

To find the common difference $d$ in an arithmetic sequence, subtract any term from the term that follows it. Mathematically, $d=\; a\_\{n+1\}\; -\; a\_n$, where $a\_n$ and $a\_\{n+1\}$ are consecutive terms. For example, if the sequence is 10, 8, 6, 4, then $d\; =\; 8\; -\; 10\; =\; -2$. This means each term decreases by 2. The common difference remains the same throughout the sequence, so you can use any two consecutive terms to find it. This value is crucial for calculating other terms and the general formula of the sequence.

What is the formula for the nth term of an arithmetic sequence?

The formula for the nth term $\mathrm{a\_n}$ of an arithmetic sequence is $\mathrm{a\_n}=\mathrm{a\_1}+\; (n\; -\; 1)d$, where $\mathrm{a\_1}$ is the first term, $d$ is the common difference, and $n$ is the term number. This formula allows you to find any term in the sequence without listing all previous terms. For example, if $\mathrm{a\_1}=\; 2$ and $d=\; 4$, then the 20th term is $\mathrm{a\_\{20\}}=\; 2\; +\; (20\; -\; 1)\; 4\; =\; 2\; +\; 76\; =\; 78$. This formula is fundamental for efficiently analyzing arithmetic sequences.

How do you find the common difference if you know the first term and another term in the sequence?

If you know the first term $\mathrm{a\_1}$ and another term $\mathrm{a\_k}$ at position $k$, you can find the common difference $d$ using the formula $\mathrm{a\_k}=\mathrm{a\_1}+\; (k\; -\; 1)d$. Rearranging to solve for $d$, you get $d=\; rac\{a\_k\; -\; a\_1\}\{k\; -\; 1\}$. For example, if $\mathrm{a\_1}=\; 2$ and $\mathrm{a\_5}=\; 14$, then $d=\; rac\{14\; -\; 2\}\{5\; -\; 1\}\; =\; rac\{12\}\{4\}\; =\; 3$. This method is useful when the common difference is not given directly.

How can you use the general term formula to find a specific term in an arithmetic sequence?

To find a specific term in an arithmetic sequence using the general term formula, plug the term number $n$ into the formula $\mathrm{a\_n}=\mathrm{a\_1}+\; (n\; -\; 1)d$. For example, if the first term $\mathrm{a\_1}$ is 2 and the common difference $d$ is 4, to find the 20th term, substitute $n\; =\; 20$: $\mathrm{a\_\{20\}}=\; 2\; +\; (20\; -\; 1)\; 4\; =\; 2\; +\; 76\; =\; 78$. This approach is faster than listing all terms and helps you calculate any term directly.