Question

Find both a function defined by a formula and a recursively defined function for the following sequences: (a) \(1,3,5,7,9,11,13, \ldots\) (b) \(1,1,3,3,5,5,7,7, \ldots\) (c) \(0,2,4,6,8, \ldots\) (d) \(1,2,4,8,16, \ldots\)

Short answer

(a) Formula: \(2n-1\), Recursive: \(a_{n}=a_{n-1}+2\). (b) Formula: \(b_n=\lceil \frac{n}{2} \rceil+\lfloor \frac{n}{2} \rfloor\), Recursive: \(b_n=b_{n-2}+2\). (c) Formula: \(2n-2\), Recursive: \(c_{n}=c_{n-1}+2\). (d) Formula: \(2^{n-1}\), Recursive: \(d_{n}=2\times d_{n-1}\).

Step-by-step solution

Analyze Sequence Pattern (a)

The sequence given is: \(1, 3, 5, 7, 9, 11, 13, \ldots\). This sequence is an arithmetic sequence where each term increases by 2. For an arithmetic sequence, each term can be expressed as \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

Formula for Sequence (a)

For sequence (a), the first term \(a_1 = 1\) and the common difference \(d = 2\). Thus, the formula using arithmetic sequence is \(a_n = 1 + (n-1) \times 2 = 2n - 1\).

Recursive Definition for Sequence (a)

A recursive definition starts with specifying the first term and giving a formula to find the next term from the previous term. For sequence (a), start with \(a_1 = 1\) and define the recursive formula as \(a_{n} = a_{n-1} + 2\).

Analyze Sequence Pattern (b)

The sequence given is: \(1, 1, 3, 3, 5, 5, 7, 7, \ldots\). This sequence increases every two terms, forming pairs of repeated values. The arithmetic sequence formed by the pattern of changes is \(1, 3, 5, 7, \ldots\) with a common difference of 2.

Formula for Sequence (b)

For sequence (b), the formula can be found using \( \frac{n+1}{2} \) for even \(n\) and \(\frac{n}{2} \) for odd \(n\). Appropriate simplifications combine these into \( b_n = \lceil \frac{n}{2} \rceil + \lfloor \frac{n}{2} \rfloor \).

Recursive Definition for Sequence (b)

For sequence (b), define \(b_1 = 1\), and for every two terms, increase by 2: i.e., \(b_n = b_{n-2} + 2\), starting from \(b_1 = 1, b_2 = 1\).

Analyze Sequence Pattern (c)

The sequence given is: \(0, 2, 4, 6, 8, \ldots\). This sequence is also an arithmetic sequence with a first term of 0 and a common difference of 2.

Formula for Sequence (c)

For sequence (c), since the common difference \(d = 2\) and the first term \(c_1 = 0\), the formula is \(c_n = 0 + (n-1) \times 2 = 2n - 2\).

Recursive Definition for Sequence (c)

For sequence (c), define the recursive formula starting with \(c_1 = 0\) and \(c_{n} = c_{n-1} + 2\).

Analyze Sequence Pattern (d)

The sequence given is \(1, 2, 4, 8, 16, \ldots\). This is a geometric sequence where each term is a power of 2, starting with 1.

Formula for Sequence (d)

For sequence (d), each term is \(d_n = 2^{n-1}\). This is because the first term is \(2^0 = 1\) and each subsequent term is multiplied by 2.

Recursive Definition for Sequence (d)

The recursive definition for sequence (d) can be defined as \(d_1 = 1\) and \(d_n = 2 \times d_{n-1}\) for \(n > 1\).

Key concepts

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is consistent. This constant difference is what sets the rhythm of the sequence, making it predictable and easy to work with.

For instance, sequence (a), which is 1, 3, 5, 7, 9, 11, 13, ..., can be noted as an arithmetic sequence with a common difference of 2. To express its general term, we use the formula:

• General formula: \( a_n = a_1 + (n-1)d \)
• Where \(a_1\) is the first term and \(d\) is the common difference.

For sequence (a), the first term \(a_1 = 1\) and the common difference \(d = 2\). By applying the arithmetic formula, you get \( a_n = 2n - 1 \). This makes it easy to find any term in the sequence without having to list out all preceding terms.

The beauty of arithmetic sequences is in their simplicity and structure, often used to model linear relationships in mathematics.

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This kind of sequence grows (or shrinks) exponentially with each term, depending on the ratio.

Looking at sequence (d) : 1, 2, 4, 8, 16, ..., we can see a pattern where each term is twice the previous term. Here, the common ratio is 2, which means the sequence is multiplying each term by 2 to get the next one. The general term is expressed with the formula:

• General formula: \( d_n = a_1 \, r^{n-1} \)
• Where \(a_1\) is the first term and \(r\) is the common ratio.

For sequence (d), the first term \(a_1 = 1\) and since the common ratio \(r = 2\), your general expression simplifies to \( d_n = 2^{n-1} \). This shows how the terms escalate quickly, a distinctive trait of geometric progressions.

These sequences are particularly useful in situations where exponential growth or decay comes into play, like in computing compound interest or population growth models.

Recursive Functions

Recursive functions express each term in a sequence based on the previous one(s). These types of definitions are useful for sequences where the relationship between consecutive terms is clear and predictable.

For example, parts of the exercise illustrate this well. For sequence (a), its recursive form is determined by:

• Start term: \( a_1 = 1 \)
• Recursive formula: \( a_{n} = a_{n-1} + 2 \)

This setup tells you how to arrive at any term based on its predecessor, detailing explicitly the addition step needed for each progression. For sequence (d), the recursive function translates to:

• Start term: \( d_1 = 1 \)
• Recursive formula: \( d_n = 2 \times d_{n-1} \)

In each case, the recursive approach focuses on defining the sequence step by step, guiding through the buildup of each term, allowing for a more intuitive understanding of the pattern.

Sequence Formulas

Working with sequences often requires a formula that allows you to determine any specific term, without the need for listing all prior terms. This is where sequence formulas become indispensable tools.

The functional formula for an arithmetic sequence provides a direct means to calculate any term. As seen with sequence (a), using \( a_n = 2n - 1 \) allows you to plug in the value of \(n\) and swiftly determine the corresponding term without manual computation of all prior ones.

For a geometric sequence, like sequence (d), the formula \( d_n = 2^{n-1} \) similarly allows rapid access to any term by incorporating the power of the common ratio. Such formulas are crucial for efficiently solving problems involving large sequences or determining specific terms quickly.

In essence, sequence formulas both simplify and clarify complex sequential patterns, reducing computational workload and highlighting inherent mathematical structures.