Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Identify arithmetic and quadratic sequences.
• Determine the general term (T_n) for both types.
• Calculate specific terms in a sequence.
• Determine the sum of an arithmetic series (S_n).
• Apply these concepts to solve problems.

Lesson summary

Number patterns are fundamental to mathematics and are observed everywhere around us, from the arrangement of tiles in a Cape Town house to the growth of savings in a bank account. Understanding number patterns allows us to predict future values, make informed decisions, and appreciate the underlying structure of the world. In the South African context, recognizing patterns in data can help us understand social trends, economic changes, and environmental issues. This week, we will focus on understanding and applying arithmetic and quadratic sequences, including finding the nth term and calculating sums of series.

Lesson notes

2.1 Arithmetic Sequences An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'.

General Form: a, a + d, a + 2d, a + 3d, ... , where 'a' is the first term.

Formula for the nth term (T n ): T n = a + (n - 1)d 'T n ' represents the nth term of the sequence. 'a' represents the first term. 'n' represents the position of the term in the sequence. 'd' represents the common difference. Formula for the sum of the first n terms (S n ): S n = n/2 [2a + (n-1)d] or S n = n/2 (a + l) where 'l' is the last term. 'S n ' represents the sum of the first n terms.

Example 1: Consider the sequence 2, 5, 8, 11, ... This is an arithmetic sequence because the difference between consecutive terms is consistently 3 (5-2 = 3, 8-5 = 3, 11-8 = 3).

Therefore, a = 2 and d =

3. Let's find the 10th term (T 10 ): T 10 = 2 + (10 - 1) 3 = 2 + 9 3 = 2 + 27 = 29 Let's find the sum of the first 10 terms (S 10 ): S 10 = 10/2 [2 2 + (10 - 1) 3] = 5 [4 + 9 3] = 5 [4 + 27] = 5 31 = 155 2.2 Quadratic Sequences A quadratic sequence is a sequence where the second difference between consecutive terms is constant. This means that the first differences form an arithmetic sequence.

General Form: T n = an 2 + bn + c, where a, b, and c are constants. 'a' cannot be zero. Finding the values of a, b, and c: Calculate the first difference (the difference between consecutive terms). Calculate the second difference (the difference between consecutive first differences). This difference is constant.

Use the following relationships: 2a = Second difference 3a + b = First difference between T 1 and T 2 a + b + c = T 1 (the first term of the original sequence) Solve these equations simultaneously to find a, b, and c.

Example 2: Consider the sequence 1, 7, 17, 31, ...

First Difference: 6, 10, 14, ...

Second Difference: 4, 4, ... (Constant) Now, we can find a, b, and c: 2a = 4 => a = 2 3a + b = 6 => 3 2 + b = 6 => 6 + b = 6 => b = 0 a + b + c = 1 => 2 + 0 + c = 1 => c = -1 Therefore, the general term is T n = 2n 2 + 0n - 1 = 2n 2 - 1 Let's find the 6th term (T 6 ): T 6 = 2(6) 2 - 1 = 2 * 36 - 1 = 72 - 1 = 71 Important Notes: Always check your general term by substituting n = 1, 2, 3... and verifying that the resulting terms match the given sequence. For arithmetic series, remember that 'd' can be positive (increasing sequence) or negative (decreasing sequence). Quadratic sequences grow faster than arithmetic sequences because of the n 2 term. Guided Practice (With Solutions)

Question 1: Determine the general term (T n ) of the arithmetic sequence: 4, 7, 10, 13, ...

Solution: Identify 'a' and 'd': a = 4, d = 7 - 4 = 3 Apply the formula: T n = a + (n - 1)d Substitute the values: T n = 4 + (n - 1) 3 Simplify: T n = 4 + 3n - 3 = 3n + 1 Therefore, the general term is T n = 3n + 1

Commentary: This question tests the basic application of the arithmetic sequence formula. Make sure to correctly identify the first term and the common difference.

Question 2: Find the 15th term of the quadratic sequence: 2, 6, 12, 20, ...

Solution: Find the first differences: 4, 6, 8, ...

Find the second differences: 2, 2, ... Calculate a, b, and c: 2a = 2 => a = 1 3a + b = 4 => 3 1 + b = 4 => b = 1 a + b + c = 2 => 1 + 1 + c = 2 => c = 0 The general term is T n = n 2 + n + 0 = n 2 + n Find T 15 : T 15 = (15) 2 + 15 = 225 + 15 = 240

Commentary: This question requires finding the general term of a quadratic sequence first. Make sure you understand the relationship between the first and second differences and the coefficients a, b, and c.

Question 3: The sum of the first 8 terms of an arithmetic series is 100, and the first term is

2. Determine the common difference (d).

Solution: We are given: S 8 = 100, a = 2, n = 8 Use the formula: S n = n/2 [2a + (n - 1)d] Substitute the values: 100 = 8/2 [2 2 + (8 - 1)d] Simplify: 100 = 4 [4 + 7d] Divide both sides by 4: 25 = 4 + 7d Subtract 4 from both sides: 21 = 7d Divide both sides by 7: d = 3

Commentary: This question involves using the sum formula in reverse to find the common difference. It requires algebraic manipulation to isolate the unknown variable 'd'. Independent Practice (Questions Only) Find the next three terms of the arithmetic sequence: -5, -1, 3, 7, ... Determine the general term of the arithmetic sequence where the 3rd term is 7 and the 7th term is

1

5. Calculate the sum of the first 20 terms of the arithmetic series: 1 + 4 + 7 + 10 + ... The general term of a quadratic sequence is given by T n = 3n 2 - 2n +

1. Find the 5th term of the sequence. Determine the general term of the quadratic sequence: 0, 3, 8, 15, ... Find the value of 'x' if the following sequence is arithmetic: x + 2, 3x - 1, 6x - 7, ... The sum of the first n terms of an arithmetic series is given by S n = n(2n + 3). Find the first term and the common difference. The first term of an arithmetic sequence is 5.