Sequences: Videos & Practice Problems

Sequences are ordered lists of numbers, where each number is termed a "term." They can be finite or infinite and often follow specific patterns, such as arithmetic or geometric progressions. Recursive formulas define terms based on previous terms, while general formulas express terms directly in terms of their index. Understanding these concepts is crucial for identifying patterns and deriving formulas, enhancing problem-solving skills in mathematics.

Introduction to Sequences

Memorizing numbers, such as bank account numbers or phone numbers, often requires understanding their order. This concept is fundamental to sequences, which are essentially lists of numbers arranged in a specific order. For instance, the sequence 2, 4, 6, 8 illustrates a clear pattern where each term increases by 2. The individual numbers in a sequence are referred to as terms, elements, or members, and they can be labeled as the first term, second term, and so on.

Sequences can exhibit patterns, making it easier to determine subsequent terms. For example, in the sequence 3, 6, 9, 12, the pattern shows that each term increases by 3. To find the fifth term, simply add 3 to the last known term (12), resulting in 15. This sequence is finite because it ends at 18. Conversely, an infinite sequence continues indefinitely, as indicated by an ellipsis (e.g., 1, 1/2, 1/3, 1/4, ...), where the pattern involves the numerator remaining constant while the denominator increases by 1.

Sequences share similarities with functions, as both follow specific rules and can be expressed with equations. However, the key difference lies in their inputs. In functions, inputs can be any real number, while sequences only accept positive integers as inputs, known as indexes, represented by the letter n. For example, in the equation \( a_n = 2n \), the outputs (terms of the sequence) are calculated by substituting integer values for n, yielding the sequence 2, 4, 6, 8, 10.

To further explore sequences, consider the formula \( a_n = n^2 \). By substituting n with 1, 2, and 3, the first three terms are calculated as follows: \( a_1 = 1^2 = 1 \), \( a_2 = 2^2 = 4 \), and \( a_3 = 3^2 = 9 \). Another example is \( a_n = \frac{1}{n + 3} \), where the first three terms are \( a_1 = \frac{1}{1 + 3} = \frac{1}{4} \), \( a_2 = \frac{1}{2 + 3} = \frac{1}{5} \), and \( a_3 = \frac{1}{3 + 3} = \frac{1}{6} \).

Sequences can also involve exponents, as seen in \( a_n = (-1)^n \). The first three terms are calculated as follows: \( a_1 = (-1)^1 = -1 \), \( a_2 = (-1)^2 = 1 \), and \( a_3 = (-1)^3 = -1 \). This sequence alternates between -1 and 1, demonstrating that sequences can include both positive and negative numbers.

In summary, sequences are ordered lists of numbers that follow specific patterns, allowing for the prediction of future terms. They can be finite or infinite and can be expressed through various mathematical formulas, making them a fundamental concept in mathematics.

The first 4 terms of a sequence are $\left\lbrace\sqrt3,2\sqrt3,3\sqrt3,4\sqrt3,\ldots\right\rbrace$. Continuing this pattern, find the $7^{\th}$ term.

• A. $8\sqrt3$
• B. $6\sqrt3$
• C. $7\sqrt3$
• D. $9\sqrt3$

Answer: (C)

Determine the first 3 terms of the sequence given by the general formula
$a_{n}=\frac{1}{n!+1}$

• A. $\left\lbrace\frac12,\frac13,\frac17\right\rbrace$
• B. $\left\lbrace\frac12,\frac13,\frac14\right\rbrace$
• C. $\left\lbrace1,2,7\right\rbrace$
• D. $\left\lbrace1,\frac12,\frac16\right\rbrace$

Answer: (A)

Writing a General Formula

In the study of sequences, understanding how to derive explicit formulas is essential. An explicit formula allows you to calculate the terms of a sequence based on their position, denoted by \( n \). The process often involves identifying patterns within the sequence of numbers.

One common pattern is an arithmetic sequence, where the terms increase by a constant amount. For example, in the sequence 5, 6, 7, 8, 9, each term increases by 1. The general formula for such sequences can be expressed as \( a_n = n + c \), where \( c \) is a constant that adjusts the starting point. In this case, since the first term is 5, we find that \( c = 4 \), leading to the formula \( a_n = n + 4 \). This can be verified by substituting values of \( n \) to ensure the correct terms are produced.

Another type of sequence involves alternating signs, such as -5, 5, -5, 5. This pattern can be represented using \( (-1)^n \), which alternates between positive and negative values based on whether \( n \) is even or odd. To adjust for the specific values in the sequence, we multiply by 5, resulting in the formula \( a_n = (-1)^n \cdot 5 \). This ensures that the first term is -5 and the second term is 5, continuing the pattern.

Sequences can also include fractions, such as \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6} \). Here, the numerators follow the pattern of \( n \), while the denominators increase by 1 starting from 2. Thus, the general formula for this sequence is \( a_n = \frac{n}{n + 1} \), which can be confirmed by substituting values of \( n \) to match the sequence.

Lastly, exponential sequences, like 2, 4, 8, 16, 32, grow at an increasing rate. The general formula for such sequences is expressed as \( a_n = b^n \), where \( b \) is the base of the exponential growth. In this case, since the sequence consists of powers of 2, the formula is \( a_n = 2^n \). This can be validated by substituting values of \( n \) to ensure the correct terms are generated.

By recognizing these patterns—arithmetic, alternating signs, fractional, and exponential—you can effectively derive explicit formulas for various sequences. This skill is crucial for solving problems related to sequences and series in mathematics.

Example 1

In this example, we explore how to derive a general formula for a sequence based on its first four terms. The sequence provided is:

1/1×2, 1/2×3, 1/3×4, 1/4×5.

To find the general term, denoted as \( a_n \), we first observe the pattern in the sequence. Notably, each term is a fraction, indicating that our general formula will also involve a fraction. The numerators of all terms are consistently 1, which suggests that the numerator in our formula will simply be 1, as it does not depend on \( n \).

Next, we analyze the denominators, which are products of two numbers: for the first term, it is \( 1 \times 2 \); for the second, \( 2 \times 3 \); for the third, \( 3 \times 4 \); and for the fourth, \( 4 \times 5 \). The first number in each denominator corresponds directly to the term's index \( n \), indicating that the first part of the denominator is \( n \).

For the second number in the denominator, we see a pattern where it increases by 1 with each term: 2, 3, 4, and 5. This suggests that the second number can be represented as \( n + 1 \). Therefore, the complete denominator can be expressed as \( n(n + 1) \).

Combining these observations, we arrive at the general formula for the sequence:

\( a_n = \frac{1}{n(n + 1)} \).

To find the 15th term of the sequence, we substitute \( n = 15 \) into our formula:

\( a_{15} = \frac{1}{15(15 + 1)} = \frac{1}{15 \times 16} = \frac{1}{240} \).

Calculating this gives approximately \( 0.0041667 \), which can be rounded to \( 0.0042 \). Thus, the 15th term in the sequence is \( 0.0042 \).

This process illustrates how to identify patterns in sequences and derive a general formula, allowing for quick calculations of specific terms without needing to list all preceding terms.

Example 2

In this exploration of sequences, we analyze a specific pattern characterized by the first five terms: -2, 4, -6, 8, -10. The goal is to derive a general formula for the nth term and subsequently calculate the 18th term. Recognizing the sequence's structure is crucial; the terms alternate in sign and increase by 2 in absolute value. This alternating pattern suggests the use of (-1) raised to the nth power, which effectively captures the sign changes. Since the first term is negative, the formula will be (-1)^n.

To account for the increasing magnitude of the terms, we observe that each term increases by 2. This can be represented by multiplying the index n by 2, leading to the term 2n. Thus, the complete formula for the nth term of the sequence can be expressed as:

a_n = (-1)ⁿ * 2n

To validate this formula, we can compute the first few terms. For n = 1, we have:

a₁ = (-1)¹ * 2(1) = -2

For n = 2:

a₂ = (-1)² * 2(2) = 4

And for n = 3:

a₃ = (-1)³ * 2(3) = -6

These calculations confirm that the formula accurately represents the sequence. To find the 18th term, we substitute n = 18 into the formula:

a₁₈ = (-1)¹⁸ * 2(18)

Since (-1)¹⁸ = 1 (as 18 is an even number), the calculation simplifies to:

a₁₈ = 1 * 36 = 36

Thus, the 18th term of the sequence is 36. This method of deriving a general formula not only streamlines the process of finding specific terms but also enhances understanding of the underlying patterns in sequences.

Recursive Formulas

In mathematics, sequences can be defined using different types of formulas, primarily general formulas and recursive formulas. A general formula provides a direct way to calculate the nth term of a sequence based on its position, n. For example, the formula \( a_n = 2n \) allows you to find the first five terms of the sequence by substituting values of n (1, 2, 3, 4, 5) to yield the outputs 2, 4, 6, 8, and 10.

On the other hand, a recursive formula defines each term based on the previous term(s) in the sequence. For instance, a recursive formula might be expressed as \( a_n = a_{n-1} + 2 \), where \( a_{n-1} \) represents the previous term. To find the terms of the sequence using this formula, you start with the first term, \( a_1 \), and use it to calculate subsequent terms. If \( a_1 = 2 \), then:

1. \( a_2 = a_1 + 2 = 2 + 2 = 4 \)

2. \( a_3 = a_2 + 2 = 4 + 2 = 6 \)

3. \( a_4 = a_3 + 2 = 6 + 2 = 8 \)

4. \( a_5 = a_4 + 2 = 8 + 2 = 10 \)

This results in the same sequence as derived from the general formula. The key distinction is that while the general formula requires knowledge of n to compute the nth term, the recursive formula relies on the previous term to find the next one.

Recursive formulas can be particularly useful when the pattern of the sequence is evident, allowing for easier calculations without needing to derive a general formula. For example, if given a recursive formula \( a_n = 2 \cdot a_{n-1} + 3 \) with \( a_1 = 1 \), the terms can be calculated as follows:

1. \( a_2 = 2 \cdot a_1 + 3 = 2 \cdot 1 + 3 = 5 \)

2. \( a_3 = 2 \cdot a_2 + 3 = 2 \cdot 5 + 3 = 13 \)

3. \( a_4 = 2 \cdot a_3 + 3 = 2 \cdot 13 + 3 = 29 \)

In this case, the sequence generated is 1, 5, 13, 29. Finding a general formula for such sequences can be complex, making recursive formulas advantageous for determining a few terms quickly.

Another example could involve a recursive formula defined as \( a_n = n \cdot a_{n-1} \) with \( a_1 = 1 \). The terms can be calculated as follows:

1. \( a_2 = 2 \cdot a_1 = 2 \cdot 1 = 2 \)

2. \( a_3 = 3 \cdot a_2 = 3 \cdot 2 = 6 \)

3. \( a_4 = 4 \cdot a_3 = 4 \cdot 6 = 24 \)

This results in the sequence 1, 2, 6, 24, showcasing how recursive formulas can effectively generate terms based on previous values. Understanding both types of formulas enhances your ability to work with sequences in various mathematical contexts.

Write the first 6 terms of the sequence given by the recursive formula $a_{n}=a_{n-2}+a_{n-1}$ ; $a_1=1$ ; $a_2=1$.

• A. $\left\lbrace1,1,2,3,5,18\right\rbrace$
• B. $\left\lbrace1,2,3,5,8,13\right\rbrace$
• C. $\left\lbrace0,1,1,2,3,5\right\rbrace$
• D. $\left\lbrace1,1,2,3,5,8\right\rbrace$

Answer: (D)

Here’s what students ask on this topic:

What is a sequence in mathematics?

A sequence in mathematics is an ordered list of numbers, where each number is called a term. The terms are arranged in a specific order, and the sequence can be either finite (having a limited number of terms) or infinite (continuing indefinitely). Sequences often follow specific patterns, such as arithmetic or geometric progressions. For example, the sequence 2, 4, 6, 8, ... is an arithmetic sequence where each term increases by 2. Understanding sequences is crucial for identifying patterns and deriving formulas in mathematics.

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

To find the nth term of an arithmetic sequence, you can use the formula:

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

where $a_{\mathrm{n}}$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference. For example, in the sequence 3, 6, 9, 12, ..., the first term $a_1$ is 3, and the common difference $d$ is 3. To find the 5th term, plug in the values: $a_5=3+(5-1)3=15$.

What is the difference between a general formula and a recursive formula in sequences?

A general formula, also known as an explicit formula, expresses the nth term of a sequence directly in terms of n. For example, in the sequence defined by $a_n=2n$, you can find any term by plugging in the value of n. A recursive formula, on the other hand, defines each term based on the previous term(s). For example, in the sequence defined by $a_n=a_{n-1}+2$ with $a_1=2$, you need to know the previous term to find the next term. General formulas are useful for finding any term directly, while recursive formulas are useful for understanding the relationship between consecutive terms.

How do you determine if a sequence is finite or infinite?

A sequence is finite if it has a limited number of terms, meaning it eventually stops. For example, the sequence 1, 2, 3, 4, 5 is finite because it ends at 5. A sequence is infinite if it continues indefinitely without stopping. For example, the sequence 1, 2, 3, 4, ... is infinite because it goes on forever. You can often identify an infinite sequence by the presence of ellipses ("...") indicating that the sequence continues. Understanding whether a sequence is finite or infinite is important for determining the scope and behavior of the sequence.

What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by $d$. For example, in the sequence 3, 6, 9, 12, ..., the common difference is 3. The general formula for the nth term of an arithmetic sequence is:

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

where $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference. Arithmetic sequences are useful in various mathematical applications, including solving problems involving linear patterns.