Sequences and Functions

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Introduction to Sequences

A sequence is an ordered list of numbers, called terms, that follow a particular pattern or rule. Sequences appear throughout mathematics and in real-world contexts such as predicting population growth, calculating compound interest, or arranging patterns of tiles, making them an important topic to master both for algebraic fluency and for practical problem solving.

Linear Sequences

A linear sequence, also called an arithmetic sequence, is one in which the difference between consecutive terms remains constant, known as the common difference. For example, the sequence 3, 7, 11, 15, 19 has a common difference of 4, since each term is found by adding 4 to the previous term. The nth term of a linear sequence can be found using the formula an + b, where a is the common difference and b is chosen so that the formula gives the correct first term.

Finding the nth Term of a Linear Sequence

To find the nth term formula of a linear sequence, first identify the common difference between consecutive terms, which becomes the coefficient of n. Then, work out what needs to be added or subtracted so that substituting n = 1 gives the correct first term. For the sequence 3, 7, 11, 15, the common difference is 4, so the formula begins with 4n; substituting n = 1 into 4n gives 4, but the first term is 3, so 1 must be subtracted, giving the nth term formula 4n - 1.

Quadratic Sequences

A quadratic sequence is one in which the second difference, that is, the difference between the differences of consecutive terms, remains constant, rather than the first difference as in a linear sequence. Quadratic sequences have an nth term formula that includes a term in n², and finding this formula involves first finding the constant second difference, halving it to find the coefficient of n², and then working out any remaining linear and constant parts.

Other Special Sequences

Beyond linear and quadratic sequences, several other important sequences appear regularly in mathematics. A geometric sequence has a constant ratio between consecutive terms, found by multiplying each term by the same value, called the common ratio, to get the next term, such as 2, 6, 18, 54 with a common ratio of 3. The Fibonacci sequence is formed by adding the two previous terms together to generate the next term, beginning 1, 1, 2, 3, 5, 8, and appears frequently in nature, such as in the arrangement of leaves and petals.

Introduction to Functions

A function is a rule that takes an input value, processes it in a defined way, and produces exactly one output value for each input. Functions are often written using function notation, such as f(x) = 2x + 3, meaning that the function f takes an input x, multiplies it by 2, and adds 3 to produce the output. Evaluating a function for a specific input, such as finding f(4), involves substituting that value in place of x throughout the function's formula.

Domain and Range

The domain of a function is the complete set of possible input values that the function can accept, while the range is the complete set of possible output values the function can produce. Some functions have restricted domains; for example, a function involving division by an expression cannot accept any input that would make the denominator equal to zero, since division by zero is undefined.

Composite Functions

A composite function is formed by applying one function to the result of another, written as fg(x), meaning that the function g is applied first to x, and then the function f is applied to the result of g(x). Composite functions must be evaluated in the correct order, working from the inside outward, since applying the functions in the wrong order will generally produce a different, incorrect result.

Inverse Functions

The inverse of a function, written f⁻¹(x), reverses the effect of the original function, so that applying a function followed by its inverse returns the original input value. Finding an inverse function typically involves writing the function as y in terms of x, swapping x and y, and then rearranging the resulting equation to make y the subject again, giving the formula for the inverse function.

Worked Example: Finding the nth Term of a Quadratic Sequence

Consider the sequence 2, 7, 14, 23, 34. The first differences are 5, 7, 9, 11, and the second differences, found by subtracting consecutive first differences, are all 2, confirming this is a quadratic sequence. Halving the constant second difference gives the coefficient of n², which is 1, so the sequence begins with n². Comparing n² (1, 4, 9, 16, 25) with the original sequence (2, 7, 14, 23, 34) shows a consistent difference of 1 added each time, giving the full nth term formula n² + 1.

Using Sequences in Real Life

Sequences model many real-world situations, such as calculating compound interest, where a sum of money grows by the same percentage each year, forming a geometric sequence, or predicting the number of seats in successive rows of a theatre that widen at a constant rate, forming a linear sequence. Recognising which type of sequence applies to a real situation is the first step in choosing the correct method to make accurate predictions.

Function Machines

A function can be visualised as a "function machine," where an input value enters the machine, is processed through one or more operations in a specific order, and an output value emerges. Function machines are a helpful way for beginners to understand how a function such as f(x) = 2x + 3 works step by step: the input x first passes through a "multiply by 2" stage, and the result then passes through an "add 3" stage before becoming the final output.

Recursive Definitions of Sequences

Some sequences are defined recursively, meaning each term is calculated directly from the previous term or terms, rather than from a general nth term formula. The Fibonacci sequence is a classic recursive example, since each term is found by adding the two preceding terms together, and recursive definitions are especially useful for sequences, such as compound interest calculations, where each new value naturally depends on the value immediately before it.

Key Terms to Remember

• Common difference: the constant amount added between consecutive terms of a linear sequence.
• nth term: a formula that generates any term of a sequence when a value of n is substituted.
• Domain: the set of all possible input values of a function.
• Composite function: a function formed by applying one function to the output of another.

Summary

Sequences and functions describe patterns and rules connecting numbers in a structured way. Linear, quadratic, geometric, and special sequences such as the Fibonacci sequence each follow their own distinctive patterns, while functions, including composite and inverse functions, provide a powerful and flexible notation for describing how one quantity depends upon another throughout mathematics.