Working with Term-to-Term Rules in Sequences - Revision Notes

In these revision notes for Working with Term-to-Term Rules in Sequences, we cover the following key points:

• What are sequences in Math?
• What are the components of sequence?
• What are the common types of sequences?
• What is sequence notation?
• How to find the nth term of a sequence in a shorter way using the sentence notation?
• What are the rules for calculating the nth terms of some common sequences through the corresponding formulas?
• How do you find the nth term of combined sequences?
• How do you find the nth term of an arithmetic sequence by solving a system of linear equations?
• What is the procedure for finding the nth term of a quadratic sequence?

Working with Term-to-Term Rules in Sequences Revision Notes

In mathematics, an ordered set of objects or numbers, like a₁, a₂, a₃, a₄, a₅, a₆, …, a_n, are said to be in a sequence, if, as per a certain rule, they have a definite value. Each of the objects or numbers contained in a sequence is called a term.

The difference between a sequence and any whatever list of items is that in sequences you are able to find the term that comes after the last term shown in the set.

As for the number of terms they contain, sequences are classified into two main types: finite and infinite. As the name itself suggests, a finite sequence contains a finite number of terms, while an infinite sequence is unlimited on one of both sides. This fact is shown by three dots preceding the first known term or succeeding the last known term of an infinite sequence.

We use the so-called "Term-to-Term Rules" to get from one term of a sequence to another.

The most common types of sequences are:

1. Arithmetic Sequences

An arithmetic sequence (otherwise known as "arithmetic progression") is a type of number sequence where the difference between two consecutive terms is always the same. Arithmetic sequences are also known as linear sequences.

The difference between two consecutive terms in an arithmetic sequence is known as the common difference, d. If the difference between two consecutive terms in an arithmetic sequence is positive the terms are increasing, otherwise, they are decreasing.

2. Geometric Sequences

A geometric sequence (otherwise known as "geometric progression") is a type of number sequence where the ratio between two consecutive terms is always the same.

The ratio between two consecutive terms in a geometric sequence is known as the common ratio, R. Like in arithmetic sequences, if the ratio between two consecutive terms in a geometric sequence is positive the terms are increasing, otherwise, they are decreasing.

3. Fibonacci-Type Sequences

A Fibonacci-type sequence is a particular kind of number sequence where the next term is obtained by adding the previous two terms. In symbols, a Fibonacci sequence is expressed as x_n = x_n - 2 + x_n - 1.

4. Quadratic Sequences

Quadratic sequences are those sequences in which the difference between two consecutive terms changes by the same amount each time when moving from left to right. Such sequences are called quadratic because they follow the pattern of quadratic polynomials/functions, where the difference between two consecutive terms changes by the same value.

5. Figure-Like Sequences

Sometimes, the patterns that certain sequences follow are similar to certain geometrical shapes such as triangles, rectangles etc. For this reason, they are called figure-like sequences.

6. Shape Sequences

Shape patterns can form sequences when they have some regularity. The most common shape sequences involve the enlargement or reduction of shapes' size according to a given rule.

The method of representing sequences by listing the first terms (known as the list method) is not very helpful when we want to find terms of a given sequence that are far from the first terms. Therefore, we often use algebra to show a sequence and its pattern, as this helps generalize the approach. In this case, we have used the sequence notation to provide information about a given sequence. For this, we denote any term of a sequence by a letter (usually x, y, u or a) and a number as an index (usually starting from 1 but sometimes the index starts from 0; however, here we will start from 1) to show the number of the term in a sequence.

We express the general term of a sequence by the index n. Hence, x_n indicates the general term (the nth term) of a sequence, x₁ indicates the first term of a sequence, x₂₇ the 27th term of that sequence, and so on.

Although the sequence notation is more suitable to represent the terms of a number sequence compared to the list method, we are not fully satisfied, as still, we have to calculate the terms one by one to get to the desired term.

In some sequences, however, it is possible to identify formulas that allow us to calculate the n^th term of a sequence (x_n) based on the first term x¹ without using the (long) recurrent procedure.

Denoting the first term of an arithmetic sequence by x₁ and the general term by x_n, we can write the following formula for the general term x_n of an arithmetic sequence after n recurring operations, in terms of the difference d and the first term x₁:

It is also possible to find the n^th term of arithmetic sequences in cases similar to the above by solving a system of linear equations. For some people, this method may seem more suitable.

We can use the same approach as above in identifying a general formula for finding the n^th term of a geometric sequence (x_n) when the first term x₁ and the common ratio R are given. The formula for the general term x_n of a geometric sequence therefore is

When a sequence is a combination of two or more types of sequences described above, the general term is calculated by considering each component sequence separately.

The n^th term of a quadratic sequence is of the type

where a, b and c are constants (a ≠ 0), while n represents the number of the term giving that the difference between consecutive terms is not constant and therefore, we cannot rely on the common difference d anymore.

The standard procedure used to find the n^th term of a quadratic sequence is:

Step 1: Find the constant second difference of the given quadratic sequence.

Step 2: Divide that number by 2 to obtain the coefficient a.

Step 3: Subtract the term an² from each known term of the quadratic expression an² + bn + c. The remaining expression is therefore linear.

Step 4: Calculate the n^th term of this linear sequence. This gives bn + c.

Step 5: Substitute all values found above in the original quadratic expression.