Section B — Chapters 23–39

Number Patterns II: General Term and Graphing

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to find the general term (n-th term) of a linear sequence.
✓By the end of this lesson students will be able to generate terms of a sequence given its general term.
✓By the end of this lesson students will be able to graph linear patterns and identify their characteristics.
✓By the end of this lesson students will be able to graph non-linear patterns and identify their characteristics.
✓By the end of this lesson students will be able to use graphs of patterns to solve problems.

Key concepts

Linear Sequence

A linear sequence (also called an arithmetic sequence) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference.

General Term (n-th term) of a Linear Sequence

The general term, denoted as T_n, is a formula that allows you to find any term in the sequence if you know its position (n). For a linear sequence, the general term can be written in the form T_n = an + b, where 'a' is the common difference and 'b' is a constant.

T_n = an + b

Non-linear Sequence

A non-linear sequence is a sequence where the difference between consecutive terms is not constant. This means the pattern does not increase or decrease by the same amount each time. Examples include sequences where terms are squared or multiplied by a constant factor.

Graphing Patterns

To graph a pattern, we plot the term number (n) on the horizontal axis (x-axis) and the value of the term (T_n) on the vertical axis (y-axis). Each term (n, T_n) is represented as a point on the graph.

Interpreting Graphs of Patterns

The graph of a linear pattern will always form a straight line. The graph of a non-linear pattern will form a curve or a series of points that do not lie on a straight line. Graphs can be used to find term values for given term numbers, or vice-versa, and to predict future terms.

Key facts to remember

1A linear sequence has a constant common difference between consecutive terms.
2The general term of a linear sequence can be written as T_n = an + b, where 'a' is the common difference.
3To find 'b' in T_n = an + b, use the first term (n=1, T_1) and solve for 'b'.
4The graph of a linear pattern is a set of points that lie on a straight line.
5The graph of a non-linear pattern is a set of points that do not lie on a straight line; they form a curve or a different shape.
6When graphing patterns, the term number (n) is plotted on the horizontal axis and the term value (T_n) is plotted on the vertical axis.

Worked examples

Example 1

Consider the linear sequence: 5, 8, 11, 14, ...\n(a) Find the common difference.\n(b) Find the general term (T_n) of the sequence.\n(c) Use the general term to find the 10th term.

I(a) To find the common difference, subtract any term from the term that follows it:

II 8 - 5 = 3

III 11 - 8 = 3

IV 14 - 11 = 3

V The common difference is 3.

VI(b) The general term for a linear sequence is T_n = an + b, where 'a' is the common difference.

VII So, T_n = 3n + b.

VIII To find 'b', substitute the first term (n=1, T_1=5) into the formula:

9 5 = 3(1) + b

10 5 = 3 + b

11 b = 5 - 3

12 b = 2

13 Therefore, the general term is T_n = 3n + 2.

14(c) To find the 10th term, substitute n=10 into the general term formula:

15 T_10 = 3(10) + 2

16 T_10 = 30 + 2

17 T_10 = 32

Answer

(a) Common difference = 3\n(b) T_n = 3n + 2\n(c) T_10 = 32

Always check your general term by substituting a few values of n to ensure it generates the correct terms of the sequence.

Example 2

A sequence is given by the general term T_n = 2n + 1.\n(a) Find the first 4 terms of the sequence.\n(b) Graph the first 4 terms of the sequence.

I(a) To find the terms, substitute n = 1, 2, 3, 4 into the general term:

II For n=1: T_1 = 2(1) + 1 = 2 + 1 = 3

III For n=2: T_2 = 2(2) + 1 = 4 + 1 = 5

IV For n=3: T_3 = 2(3) + 1 = 6 + 1 = 7

V For n=4: T_4 = 2(4) + 1 = 8 + 1 = 9

VI The first 4 terms are 3, 5, 7, 9.

VII(b) The points to plot are (n, T_n): (1, 3), (2, 5), (3, 7), (4, 9).

VIII Plot these points on a coordinate plane with 'n' on the horizontal axis and 'T_n' on the vertical axis.

9 Since it is a linear sequence, the points will lie on a straight line. You can draw a straight line through these points.

Answer

(a) The first 4 terms are 3, 5, 7, 9.\n(b) Graph shows points (1,3), (2,5), (3,7), (4,9) lying on a straight line.

When graphing, ensure your axes are clearly labelled and scaled appropriately.

Example 3

Consider the sequence given by T_n = n^2 + 1.\n(a) Find the first 4 terms of the sequence.\n(b) Graph the first 4 terms of the sequence and state whether it is linear or non-linear.

I(a) To find the terms, substitute n = 1, 2, 3, 4 into the general term:

II For n=1: T_1 = (1)^2 + 1 = 1 + 1 = 2

III For n=2: T_2 = (2)^2 + 1 = 4 + 1 = 5

IV For n=3: T_3 = (3)^2 + 1 = 9 + 1 = 10

V For n=4: T_4 = (4)^2 + 1 = 16 + 1 = 17

VI The first 4 terms are 2, 5, 10, 17.

VII(b) The points to plot are (n, T_n): (1, 2), (2, 5), (3, 10), (4, 17).

VIII Plot these points on a coordinate plane.

9 Observe the pattern of the points: The differences between consecutive terms are 3, 5, 7. Since the differences are not constant, the sequence is non-linear. The points on the graph will not form a straight line; they will form a curve.

Answer

(a) The first 4 terms are 2, 5, 10, 17.\n(b) Graph shows points (1,2), (2,5), (3,10), (4,17) forming a curve. The sequence is non-linear.

For non-linear patterns, do not connect the points with a straight line. A smooth curve can be drawn if the pattern is clearly defined, but for 1st year, plotting discrete points is sufficient.

Common mistakes

✗Confusing the term number (n) with the value of the term (T_n).
✗Incorrectly calculating the common difference, leading to an incorrect general term.
✗Assuming all patterns are linear and trying to fit a straight line to non-linear data.
✗Plotting points incorrectly on the graph, or mislabelling the axes.
✗Connecting points with a straight line for non-linear patterns.

Exam tips

★Always show your full step-by-step working when finding the general term or generating terms.
★Use graph paper for all graphing questions and ensure your axes are clearly labelled with 'n' and 'T_n' (or appropriate variables) and have a suitable scale.
★Check your general term by substituting the first few values of 'n' to see if it produces the correct terms of the sequence.
★For graph-based problems, read values carefully from the graph and indicate on the graph how you obtained your answer (e.g., by drawing lines to the axes).

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