Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Identify arithmetic (linear) number patterns.
• Determine the common difference in a sequence.
• Derive and use the general formula for an arithmetic sequence.
• Find any term in a sequence using the formula.
• Solve real-world problems involving these sequences.

Lesson summary

Number patterns are all around us, from the arrangement of seeds in a sunflower to the growth of our national debt. Understanding number patterns is a crucial skill in mathematics that provides a foundation for more advanced topics like sequences and series, calculus, and even financial modelling. In a South African context, recognizing and understanding patterns can help us analyse trends in population growth, predict resource consumption, and even understand the spread of diseases like HIV/AIDS or analyze economic trends such as inflation rates.

Lesson notes

An arithmetic (linear) number pattern (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, denoted by d. The first term of the sequence is usually denoted by a or T

1. General Formula for Arithmetic Sequences: The general formula for the nth term (Tn) of an arithmetic sequence is: Tn = a + (n - 1)d Where: Tn is the n*th term of the sequence (the term we want to find). a is the first term of the sequence. n is the position of the term in the sequence (e.g., 1st, 2nd, 3rd, ...). d is the common difference between consecutive terms. Why does this formula work?

Let's consider a sequence: 2, 5, 8, 11, 14, ... The first term (a) is

2. The common difference (d) is 3 (5-2 = 3, 8-5 = 3, etc.). To get to the second term (5), we add the common difference (3) once to the first term (2). To get to the third term (8), we add the common difference (3) twice to the first term (2). To get to the fourth term (11), we add the common difference (3) three times to the first term (2).

Therefore, to get to the nth term, we add the common difference (n-1) times to the first term. This is precisely what the formula T_n = a + (n-1)d represents.

Worked example

Example 1:

Consider the sequence: 4, 7, 10, 13, ...

a) Determine the common difference (d).

b) Find the general formula (Tn).

c) Calculate the 20th term (T20).

Solution: