Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Identify quadratic number patterns.
• Determine the general term (Tn) of a quadratic pattern.
• Use the general term to find any specific term.
• Apply quadratic patterns to solve problems.

Lesson summary

Number patterns are a fundamental part of mathematics and are essential for developing logical reasoning, problem-solving skills, and the ability to make predictions. In South Africa, understanding number patterns can help us analyse trends in various sectors, from population growth to economic indicators, and even predict future occurrences such as resource demand. Recognizing and applying patterns is a vital skill needed in many careers, including finance, engineering, and computer science, contributing to the country’s growth and development. This week, we will focus on identifying, describing, and extending quadratic number patterns.

Lesson notes

What is a Quadratic Number Pattern? A quadratic number pattern is a sequence of numbers where the second difference between consecutive terms is constant. This means that if you subtract each term from the term that follows it, you will get a new sequence of numbers. If you then find the difference between consecutive terms in that sequence, you will get a constant value.

Let's illustrate this: Consider the sequence: 2, 5, 10, 17, 26...

First Difference: 5-2 = 3; 10-5 = 5; 17-10 = 7; 26-17 = 9 Second Difference: 5-3 = 2; 7-5 = 2; 9-7 = 2 Since the second difference is constant (2 in this case), this is a quadratic number pattern. The General Term (Tn) of a Quadratic Number Pattern The general term of a quadratic number pattern is given by the formula: `Tn = an^2 + bn + c` where: `Tn` represents the nth term in the sequence. `n` represents the position of the term in the sequence (e.g., n=1 for the first term, n=2 for the second term, and so on). `a`, `b`, and `c` are constants that need to be determined for each specific quadratic pattern. Finding the values of a, b, and c To find the values of `a`, `b`, and `c`, we use the following relationships derived from the first three terms of the sequence: 2a = Second Difference 3a + b = First difference between T1 and T2 a + b + c = T1 (the first term in the sequence)

Let's apply this to the example above: 2, 5, 10, 17, 26... 2a = 2 => a = 1 3a + b = 3 (since 5-2 = 3) => 3(1) + b = 3 => b = 0 a + b + c = 2 => 1 + 0 + c = 2 => c = 1 Therefore, the general term for this sequence is: Tn = 1n^2 + 0n + 1 = n^2 + 1 Example 1: Find the general term of the quadratic sequence: 3, 6, 11, 18, 27… First Difference: 6-3 = 3; 11-6 = 5; 18-11 = 7; 27-18 = 9 Second Difference: 5-3 = 2; 7-5 = 2; 9-7 = 2 2a = 2 => a = 1 3a + b = 3 => 3(1) + b = 3 => b = 0 a + b + c = 3 => 1 + 0 + c = 3 => c = 2 Therefore, the general term is: Tn = n^2 + 2 Example 2: Determine the 15th term of the sequence: 1, 7, 17, 31… First Difference: 7-1 = 6; 17-7 = 10; 31-17 = 14 Second Difference: 10-6 = 4; 14-10 = 4 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: Tn = 2n^2 - 1 To find the 15th term (T15), substitute n = 15 into the general term: T15 = 2(15)^2 - 1 = 2(225) - 1 = 450 - 1 = 449 Example 3: A pattern is formed by matchsticks. The number of matchsticks needed for the first three figures are 7, 15, and 25 respectively. Determine the number of matchsticks needed for the 6th figure.

The sequence is: 7, 15, 25,...

First Difference: 15-7 = 8; 25-15 = 10 Second Difference: 10-8 = 2 2a = 2 => a = 1 3a + b = 8 => 3(1) + b = 8 => 3 + b = 8 => b = 5 a + b + c = 7 => 1 + 5 + c = 7 => 6 + c = 7 => c = 1 Therefore, the general term is: Tn = n^2 + 5n + 1 To find the number of matchsticks needed for the 6th figure (T6), substitute n = 6 into the general term: T6 = (6)^2 + 5(6) + 1 = 36 + 30 + 1 = 67 Guided Practice (With Solutions)

Question 1: Identify whether the sequence 4, 9, 16, 25, 36 is a quadratic sequence.

Solution: First Difference: 9 - 4 = 5; 16 - 9 = 7; 25 - 16 = 9; 36 - 25 = 11 Second Difference: 7 - 5 = 2; 9 - 7 = 2; 11 - 9 = 2 Since the second difference is constant (2), the sequence is quadratic.

Question 2: Determine the general term (Tn) of the following sequence: 0, 3, 8, 15… Solution: First Difference: 3 - 0 = 3; 8 - 3 = 5; 15 - 8 = 7 Second Difference: 5 - 3 = 2; 7 - 5 = 2 2a = 2 => a = 1 3a + b = 3 => 3(1) + b = 3 => b = 0 a + b + c = 0 => 1 + 0 + c = 0 => c = -1 Therefore, the general term is: Tn = n^2 - 1 Question 3: Find the 8th term in the quadratic sequence where T(n) = 3n^2 - 2n + 1 Solution: We are given Tn = 3n^2 - 2n +

1. We want to find T

8. Substituting n = 8: T8 = 3(8)^2 - 2(8) + 1 = 3(64) - 16 + 1 = 192 - 16 + 1 = 177 Question 4: The first three terms of a quadratic sequence are 2, x, and 2x +

1. Find the value of x.

Solution: The first difference between T1 and T2 is x -

2. The first difference between T2 and T3 is (2x + 1) - x = x +

1. Since it's a quadratic sequence, the second difference is constant.

Therefore: (x + 1) - (x - 2) = constant x + 1 - x + 2 = constant 3 = second difference Since 2a = second difference: 2a = 3, and a = 1.5 Now, using the fact that the second difference is constant, we also know (x+1) - (x-2) must be equal to x - 2 - 2 (the "first" first difference, if we went one step back).

That gives: x+1 - (x-2) = 3 = x - 2 -2. 3 = x -4 x = 7 We can check: The sequence is 2, 7,

1

5. First differences: 5,

8. Second difference:

3. This agrees.

Question 5: The number of chairs in rows in a hall forms the following pattern: 5, 8, 13....How many chairs would be in the 7th row?

Solution: The sequence is: 5, 8, 13,...