Algebra Study Guide

When we talk about 'special sequences' in the context of graphs, we are usually referring to sequences that produce recognisable patterns when their terms are plotted against their position in the sequence. For example, a linear sequence is an arithmetic progression where the difference between consecutive terms is constant. When plotted, these points lie on a straight line. The position of the term (e.g., 1st, 2nd, 3rd) is typically plotted on the x-axis, and the value of the term itself is plotted on the y-axis.

A key aspect of graphing sequences is to understand that the input values (the term positions) are always integers, usually starting from 1. This means that when you plot a sequence, you'll get a series of distinct points, not a continuous line. However, we often draw a line through these points to visualise the trend or the rule generating the sequence. For linear sequences, the rule is typically of the form T(n) = an + b, where 'a' is the common difference and 'b' is a constant. The graph will be a set of points (n, T(n)).

Other 'special sequences' might include quadratic sequences, where the second difference is constant. These sequences, when plotted, would show points that lie on a curve resembling a parabola. For example, the terms of a sequence given by T(n) = n squared would produce points (1,1), (2,4), (3,9), etc., which clearly form part of a parabolic curve. The ability to recognise these visual patterns helps in identifying the type of sequence and potentially its generating rule. It also involves understanding how to read information from these graphs, such as finding the value of a term given its position, or vice-versa.

Special sequences are ordered lists of numbers that follow a specific pattern or rule. In GCSE Mathematics, you'll primarily encounter three types: arithmetic, geometric, and quadratic sequences. Understanding their properties is crucial, especially when relating them to graphs.

An arithmetic sequence has a common difference (d) between consecutive terms. For example, 2, 5, 8, 11... has d = 3. The general term can be written as a_n = a_1 + (n-1)d, where a_1 is the first term and n is the term number. When plotted on a graph with term number (n) on the x-axis and term value (a_n) on the y-axis, an arithmetic sequence forms a straight line. This directly relates to linear functions, y = mx + c, where the common difference 'd' is equivalent to the gradient 'm'.

A geometric sequence has a common ratio (r) between consecutive terms. For example, 2, 6, 18, 54... has r = 3. The general term is a_n = a_1 * r^(n-1). When plotted on a graph, a geometric sequence does NOT form a straight line. Instead, it shows exponential growth or decay, depending on the common ratio. If r > 1 or r < -1, it will grow exponentially; if -1 < r < 1 (and r ≠ 0), it will decay exponentially. If r=1 or r=-1, it's a special case.

A quadratic sequence has a constant second difference. This means that if you find the differences between consecutive terms (the first differences), and then find the differences between those first differences (the second differences), the second differences will be constant. For example, 1, 4, 9, 16, 25... is a quadratic sequence (the squares of integers). The general term can be written in the form a_n = an^2 + bn + c. When plotted on a graph, a quadratic sequence forms a curve, specifically a parabola, which is characteristic of quadratic functions, y = ax^2 + bx + c.

Being able to identify these sequences from a list of numbers or from their graphical representation, and to determine their rules, is a key skill. This involves calculating common differences, common ratios, or finding the constant second difference, and then using these to derive the nth term formula.