Arithmetic Sequences: Videos & Practice Problems

Arithmetic sequences are characterized by a constant difference, known as the common difference (d), between consecutive terms. A recursive formula for an arithmetic sequence can be expressed as an = an-1+d, where the first term is specified. The general formula for the nth term is an = a1+d(n-1), allowing for efficient term calculation without prior terms. Understanding these formulas is essential for analyzing sequences effectively.

Arithmetic Sequences - Recursive Formula

Arithmetic sequences are a specific type of sequence where the difference between consecutive terms remains constant. This constant difference is referred to as the common difference, denoted by the letter d. For example, in the sequence 2, 7, 12, 17, each term increases by 5, making the common difference d = 5.

To express an arithmetic sequence using a recursive formula, we can define the next term based on the previous term. The general form of the recursive formula is:

\( a_n = a_{n-1} + d \)

Here, \( a_n \) represents the nth term, \( a_{n-1} \) is the previous term, and d is the common difference. For the sequence mentioned, the recursive formula would be:

\( a_n = a_{n-1} + 5 \)

To fully define the sequence, it is essential to specify the first term. In this case, the first term \( a_1 \) is 2.

For example, if we start with \( a_1 = 3 \) and have a common difference of 4, we can calculate the first four terms as follows:

• \( a_1 = 3 \)
• \( a_2 = a_1 + 4 = 3 + 4 = 7 \)
• \( a_3 = a_2 + 4 = 7 + 4 = 11 \)
• \( a_4 = a_3 + 4 = 11 + 4 = 15 \)

Thus, the first four terms are 3, 7, 11, and 15.

Conversely, if the common difference is negative, such as in a sequence starting with \( a_1 = 9 \) and a common difference of -6, the terms would be calculated as follows:

• \( a_1 = 9 \)
• \( a_2 = a_1 - 6 = 9 - 6 = 3 \)
• \( a_3 = a_2 - 6 = 3 - 6 = -3 \)
• \( a_4 = a_3 - 6 = -3 - 6 = -9 \)

In this case, the first four terms are 9, 3, -3, and -9.

When tasked with writing a recursive formula from a given sequence, the first step is to determine the common difference by subtracting any two consecutive terms. For instance, in the sequence 2, 5, 8, 11, the common difference is:

\( d = 5 - 2 = 3 \)

Thus, the recursive formula can be established as:

\( a_n = a_{n-1} + 3 \)

Finally, it is crucial to specify the first term, which in this case is \( a_1 = 2 \). This ensures that the recursive formula is complete and can be used to generate any term in the sequence.

Write a recursive formula for the arithmetic sequence.
$\left\lbrace8,2,-4,-10,\ldots\right\rbrace$

• A. $a_{n}=a_{n-1}-10$ ; $a_1=6$
• B. $a_{n}=a_{n-1}-10$ ; $a_1=6$
• C. $a_{n}=a_{n-1}-6$ ; $a_1=8$
• D. $a_{n}=a_{n-1}-10$ ; $a_1=8$

Answer: (C)

Arithmetic Sequences - General Formula

In arithmetic sequences, understanding how to derive a general formula is essential for efficiently calculating terms without relying on previous values. A recursive formula, which involves the previous term, can be cumbersome for high indices, such as the 101st term. Instead, a general formula allows for direct computation of any term in the sequence.

The general formula for the nth term of an arithmetic sequence is expressed as:

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

In this formula, \( a_n \) represents the nth term, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term's position in the sequence.

To illustrate, consider the sequence 2, 7, 12, 17. Here, the first term \( a_1 \) is 2, and the common difference \( d \) is 5. Plugging these values into the general formula gives:

\( a_n = 2 + 5(n - 1) \)

To find the 4th term, substitute \( n = 4 \):

\( a_4 = 2 + 5(4 - 1) = 2 + 15 = 17 \)

This confirms that the 4th term is indeed 17, consistent with the sequence.

Now, let’s apply this to another sequence: 2, 5, 8, 11, 14. The first term \( a_1 \) is 2, and the common difference \( d \) is 3. The general formula becomes:

\( a_n = 2 + 3(n - 1) \)

To find the 101st term, substitute \( n = 101 \):

\( a_{101} = 2 + 3(101 - 1) = 2 + 3 \times 100 = 2 + 300 = 302 \)

This method demonstrates the power of general formulas in arithmetic sequences, allowing for quick calculations without the need for extensive prior computations. Understanding and applying this formula is crucial for efficiently solving problems involving sequences.

Find the general formula for the arithmetic sequence below. Without using a recursive formula, calculate the $30^{\th}$ term.
$\left\lbrace-9,-4,1,6,\ldots\right\rbrace$

• A. 46
• B. 136
• C. 146
• D. 150

Answer: (B)

Example 1

In mathematics, particularly in sequences, converting a recursive formula into a general formula can significantly simplify calculations. For an arithmetic sequence, where each term is derived from the previous term by adding a constant, this process is straightforward. In this example, we start with a recursive formula where the next term is the previous term plus 3, with a starting term of 2.

The general formula for an arithmetic sequence can be expressed as:

a_n = a₁ + d(n - 1)

Here, a_n represents the nth term, a₁ is the first term, d is the common difference, and n is the term index. In this case, the first term a₁ is 2, and the common difference d is 3. Thus, the general formula becomes:

a_n = 2 + 3(n - 1)

This formula allows us to calculate any term in the sequence without needing to derive all previous terms recursively. For instance, to find the 15th term, we substitute n with 15:

a₁₅ = 2 + 3(15 - 1)

Calculating this gives:

a₁₅ = 2 + 3(14) = 2 + 42 = 44

Therefore, the 15th term of the sequence is 44. This method of deriving a general formula from a recursive one not only streamlines the process but also enhances understanding of the sequence's structure.

Example 2

In this problem, we are tasked with finding the 18th term of an arithmetic sequence given the 4th term \( a_4 = -2 \) and the 6th term \( a_6 = 6 \). To approach this, we start with the general formula for the \( n \)-th term of an arithmetic sequence, which is expressed as:

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

Here, \( a_1 \) represents the first term, and \( d \) is the common difference. Since we only have the values for the 4th and 6th terms, we can set up two equations based on the general formula:

1. For \( n = 4 \): \( a_4 = a_1 + 3d = -2 \)

2. For \( n = 6 \): \( a_6 = a_1 + 5d = 6 \)

These equations can be rewritten as:

1. \( a_1 + 3d = -2 \)

2. \( a_1 + 5d = 6 \)

Next, we can solve this system of equations using the elimination method. By subtracting the first equation from the second, we eliminate \( a_1 \):

\( (a_1 + 5d) - (a_1 + 3d) = 6 - (-2) \)

This simplifies to:

\( 2d = 8 \)

From which we find:

\( d = 4 \)

Now that we have the common difference \( d \), we can substitute it back into one of the original equations to find \( a_1 \). Using the first equation:

\( a_1 + 3(4) = -2 \)

This simplifies to:

\( a_1 + 12 = -2 \)

Thus, we find:

\( a_1 = -14 \)

With both \( a_1 \) and \( d \) known, we can now write the general formula for the sequence:

\( a_n = -14 + 4(n - 1) \)

To find the 18th term, we substitute \( n = 18 \) into the formula:

\( a_{18} = -14 + 4(18 - 1) \)

This simplifies to:

\( a_{18} = -14 + 4(17) \)

Calculating further:

\( a_{18} = -14 + 68 = 54 \)

Therefore, the 18th term of the sequence is \( 54 \).

Here’s what students ask on this topic:

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

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted by $d$. To identify an arithmetic sequence, check if the difference between each pair of consecutive terms is the same. For example, in the sequence 2, 5, 8, 11, the common difference is 3 because each term increases by 3.

How do you write a recursive formula for an arithmetic sequence?

To write a recursive formula for an arithmetic sequence, use the format $a_n=a_{\mathrm{n-1}}+d$, where $a_n$ is the nth term, $a_{\mathrm{n-1}}$ is the previous term, and $d$ is the common difference. Additionally, specify the first term $a_1$. For example, for the sequence 3, 7, 11, 15 with a common difference of 4, the recursive formula is $a_n=a_{\mathrm{n-1}}+\; 4$ with $a_1=\; 3$.

How do you find the nth term of an arithmetic sequence using the general formula?

To find the nth term of an arithmetic sequence, use the general formula $a_n=a_1+d(n-\; 1)$, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number. For example, in the sequence 2, 7, 12, 17 with $a_1=\; 2$ and $d=\; 5$, the 4th term is calculated as $a_4=\; 2\; +\; 5(4\; -\; 1)\; =\; 2\; +\; 15\; =\; 17$.

What is the common difference in an arithmetic sequence and how do you find it?

The common difference in an arithmetic sequence is the constant amount by which each term increases or decreases. It is denoted by $d$. To find the common difference, subtract any term from the subsequent term. For example, in the sequence 4, 9, 14, 19, the common difference is $9\; -\; 4\; =\; 5$ or $14\; -\; 9\; =\; 5$. The common difference remains the same throughout the sequence.

How do you determine if a sequence is arithmetic?

To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. This difference is called the common difference, $d$. For example, in the sequence 10, 15, 20, 25, the difference between each pair of consecutive terms is $15\; -\; 10\; =\; 5$, $20\; -\; 15\; =\; 5$, and $25\; -\; 20\; =\; 5$. Since the difference is constant, the sequence is arithmetic.