Algebra Revision Notes

Using an nth Term Formula to Generate Terms

An nth term formula, often denoted as T_n or a_n, provides a rule to calculate any term of a sequence directly from its position, 'n'. To find a specific term, substitute the term's position into the formula. For example, if T_n = n^2 - 2n + 5, to find the 4th term, replace 'n' with '4': T_4 = (4)^2 - 2(4) + 5 = 16 - 8 + 5 = 13. This method is efficient for finding terms far along in the sequence without needing to calculate all preceding terms. Ensure careful substitution and calculation, especially with negative numbers or fractions.

Generating Terms from Recurrence Relations

A recurrence relation defines a term of a sequence based on one or more previous terms, along with one or more initial terms. For instance, if a_{n+1} = 3a_n - 2 with a_1 = 5. To find the second term, a_2, use a_1: a_2 = 3a_1 - 2 = 3(5) - 2 = 15 - 2 = 13. For the third term, a_3, use a_2: a_3 = 3a_2 - 2 = 3(13) - 2 = 39 - 2 = 37. Always start with the given initial term(s) and work your way up. Pay close attention to subscripts to identify which term is being defined and which terms are being used.

Recognising and Applying Different Types of Rules

Sequences can follow various patterns. Beyond simple arithmetic (constant difference) and geometric (constant ratio) sequences, you will encounter quadratic sequences (where the nth term involves n^2) and more complex non-linear forms. When given a rule, whether it's an nth term formula or a recurrence relation, identify the type of operation (addition, subtraction, multiplication, division, powers) and the variables involved. For example, T_n = n^3 - 1 involves cubing the term number, while T_{n+1} = T_n / 2 involves division. Accurately interpreting the rule is the first step to correctly generating the sequence.

Common Pitfalls and Precision

A common mistake is incorrect substitution, especially with n^2 or (n-1) terms, where the order of operations matters. Be careful with signs when substituting negative values or when multiplying. When generating terms, always write down the intermediate steps to minimise errors. If a recurrence relation uses more than one previous term (e.g., T_{n+2} = T_{n+1} + T_n), ensure all required initial terms are provided and correctly used. Precision in calculations is key, as a single error early in a recurrence relation will propagate through subsequent terms.

Plotting Linear Graphs

A linear graph is a straight line formed by an equation in the form of y = mx + c, where 'm' is the gradient and 'c' is the y-intercept. To plot a linear graph, you can choose at least two (but preferably three or more) x-values, substitute them into the equation to find the corresponding y-values, create a table of coordinates (x, y), and then plot these points on a coordinate grid. Once plotted, draw a straight line through all the points, extending it across the grid. For example, for y = 2x + 1, if x = 0, y = 1; if x = 1, y = 3; if x = 2, y = 5. Points are (0,1), (1,3), (2,5).

Plotting Non-Linear Graphs (e.g., Quadratics)

Non-linear graphs represent equations that do not form a straight line, such as quadratic equations like y = x² + 2x - 3. To plot these graphs, you need to calculate more points than for a linear graph to accurately capture the curve's shape. Typically, you might calculate y-values for x-values ranging from -3 to 3 or -4 to 4. Create a table of x and y values, plot each (x, y) coordinate, and then join them with a smooth curve. Do not use a ruler for non-linear graphs. The curve for a quadratic equation is called a parabola and is symmetrical.

Identifying Points on a Graph

Given a graph, you can read off the coordinates of any point on the line or curve. The x-coordinate is the horizontal position, and the y-coordinate is the vertical position. You can also check if a specific point lies on a graph by substituting its x and y coordinates into the equation. If the equation holds true, the point is on the graph. For example, for y = 3x - 2, the point (2, 4) is on the line because substituting x=2 gives y = 3(2) - 2 = 6 - 2 = 4.

Generating Sequences from Rules

A sequence is a list of numbers following a specific rule. The rule often defines the nth term of the sequence, denoted as T_n. To generate terms of a sequence, substitute the position number (n = 1, 2, 3, ...) into the rule. For example, if the nth term is given by T_n = 2n + 3: for the 1st term (n=1), T_1 = 2(1) + 3 = 5; for the 2nd term (n=2), T_2 = 2(2) + 3 = 7; for the 3rd term (n=3), T_3 = 2(3) + 3 = 9. The sequence is 5, 7, 9, ... These terms can also be plotted as discrete points (n, T_n) on a graph.

What are Special Sequences on Graphs?

Special sequences, when plotted on a graph, reveal specific patterns. The most common type encountered at Foundation level is the linear sequence (also known as an arithmetic progression). In a linear sequence, the difference between consecutive terms is constant. When you plot the term number (n) on the x-axis and the term value (T(n)) on the y-axis, the points will lie on a straight line. For example, the sequence 3, 5, 7, 9... has a common difference of 2. Plotting (1,3), (2,5), (3,7), (4,9) will show points on a straight line. It's important to remember that these are discrete points, as 'n' (the term number) can only be a positive integer (like 1, 2, 3, ...), not a fraction or decimal.

Plotting Linear Sequences

To plot a linear sequence, first identify the term numbers (n = 1, 2, 3, ...) and their corresponding term values (T(n)). Create a table of values: n | T(n). For instance, if the nth term rule is T(n) = 2n + 1, then for n=1, T(1) = 2(1)+1 = 3; for n=2, T(2) = 2(2)+1 = 5; for n=3, T(3) = 2(3)+1 = 7. The points you would plot are (1,3), (2,5), (3,7). Always label your axes clearly: 'Term Number' or 'n' for the x-axis and 'Term Value' or 'T(n)' for the y-axis. Choose a sensible scale for both axes to ensure all your points fit and the graph is easy to read. While points on a graph of a sequence should typically *not* be connected, if a trend line is requested or implied, it will pass through these points.

Recognising Patterns and Rules from Graphs

If you are given a graph with points plotted for a sequence, you can determine if it's a linear sequence by checking if the points form a straight line. If they do, you can find the rule. The common difference of the sequence corresponds to the gradient of the line. The 'zeroth term' (the value when n=0, obtained by extending the line backwards) helps in finding the constant part of the rule. For example, if the points are (1, 4), (2, 7), (3, 10), the common difference is 3. Extending backward, for n=0, the value would be 4-3=1. So the rule is T(n) = 3n + 1. Graphs can also show non-linear sequences; for example, points forming a U-shape would suggest a quadratic sequence.

Identifying Sequence Types Graphically

When terms of a sequence are plotted against their term number (n), the shape of the graph reveals the type of sequence.

• Arithmetic sequences produce a straight line graph. The common difference 'd' corresponds to the gradient of this line. For example, if d=3, the line will have a gradient of 3.
• Geometric sequences produce an exponential curve. This curve will show rapid growth (if the common ratio 'r' is > 1) or decay (if 0 < r < 1).
• Quadratic sequences produce a parabolic curve. This characteristic 'U' or 'n' shape indicates a quadratic relationship between term number and term value.

nth Term Rules and Their Graphical Link

The general rules for the nth term of these sequences have direct graphical interpretations:

• Arithmetic: a_n = a_1 + (n-1)d or a_n = dn + c. This is a linear function y = mx + c where m=d and c = a_1 - d. The graph is a straight line.
• Geometric: a_n = a_1 * r^(n-1). This is an exponential function. The graph is an exponential curve.
• Quadratic: a_n = an^2 + bn + c. This is a quadratic function y = ax^2 + bx + c. The graph is a parabola.

Understanding these forms helps in both deriving the rule and predicting the graph shape.

Calculating Differences and Ratios

To determine the type of sequence and its rule:

• Arithmetic: Calculate the differences between consecutive terms. If these 'first differences' are constant, it's arithmetic. This constant value is 'd'.
• Geometric: Calculate the ratio between consecutive terms (Term_n / Term_(n-1)). If these 'ratios' are constant, it's geometric. This constant value is 'r'.
• Quadratic: If the first differences are not constant, calculate the differences between the first differences. If these 'second differences' are constant, it's quadratic. This constant value is 2a in the an^2 + bn + c form.