Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Determine the general term (Tn) of a quadratic sequence.
• Calculate the sum (Sn) of an arithmetic series.
• Solve problems involving these sequences and series.
• Differentiate between arithmetic and quadratic sequences.

Lesson summary

Number patterns are a fundamental part of mathematics that helps us understand the world around us. Identifying and working with patterns helps develop problem-solving skills, logical thinking, and the ability to make predictions. In South Africa, understanding number patterns can be applied to various real-life situations, from budgeting household expenses based on predictable income cycles to analyzing trends in population growth or economic data. This week we will focus on quadratic sequences and arithmetic series, extending our knowledge from Grade 10.

Lesson notes

A. Quadratic Sequences A quadratic sequence is a sequence of numbers where the second difference between consecutive terms is constant. This differs from arithmetic sequences, where the first difference is constant. The general form of a quadratic sequence is: `Tn = an² + bn + c` where 'a', 'b', and 'c' are constants, and 'n' represents the term number (n ∈ N). How to Find a, b, and c: Calculate the first difference: Subtract each term from the term that follows it.

Calculate the second difference: Subtract each first difference from the first difference that follows it. If the second difference is constant, the sequence is quadratic.

Use the following equations: `2a = Second difference` `3a + b = First difference between T1 and T2` `a + b + c = T1` Example 1: Consider the sequence: 2, 7, 14, 23, 34, ...

First difference: 7-2=5, 14-7=7, 23-14=9, 34-23=11 Second difference: 7-5=2, 9-7=2, 11-9=2 (Constant!) Finding a, b, and c: `2a = 2 => a = 1` `3a + b = 5 => 3(1) + b = 5 => b = 2` `a + b + c = 2 => 1 + 2 + c = 2 => c = -1` Therefore, the general term is: `Tn = n² + 2n - 1` Verification: Let's check if T3 is correct. T3 = (3)² + 2(3) - 1 = 9 + 6 - 1 =

1

4. This matches the given sequence.

Example 2: Find the general term of the sequence: 6, 15, 28, 45, 66,...

First difference: 9, 13, 17, 21 Second difference: 4, 4, 4 (Constant!) Finding a, b, and c: `2a = 4 => a = 2` `3a + b = 9 => 3(2) + b = 9 => b = 3` `a + b + c = 6 => 2 + 3 + c = 6 => c = 1` Therefore, the general term is: `Tn = 2n² + 3n + 1` B. Arithmetic Series An arithmetic series is the sum of the terms in an arithmetic sequence. An arithmetic sequence has a constant difference between consecutive terms (common difference, denoted by 'd'). The sum of the first 'n' terms of an arithmetic series is given by: `Sn = n/2 [2a + (n-1)d]` or `Sn = n/2 [a + l]` Where: `Sn` = Sum of the first 'n' terms. `n` = Number of terms. `a` = First term of the sequence. `d` = Common difference. `l` = Last term of the sequence (Tn).

Example 3: Find the sum of the first 20 terms of the arithmetic series: 3 + 7 + 11 + 15 + ...

Identify a and d: `a = 3` `d = 7 - 3 = 4` Apply the formula: `S20 = 20/2 [2(3) + (20-1)4]` `S20 = 10 [6 + (19)4]` `S20 = 10 [6 + 76]` `S20 = 10 [82]` `S20 = 820` Example 4: The sum of the first 'n' terms of an arithmetic series is given by Sn = n(3n + 1). Find the first three terms of the series. Find S1, S2, and S3: `S1 = 1(3(1) + 1) = 1(4) = 4` `S2 = 2(3(2) + 1) = 2(7) = 14` `S3 = 3(3(3) + 1) = 3(10) = 30` Find the terms: `T1 = S1 = 4` `T2 = S2 - S1 = 14 - 4 = 10` `T3 = S3 - S2 = 30 - 14 = 16` Therefore, the first three terms are 4, 10, and

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6. Guided Practice (With Solutions)

Question 1: Find the general term of the quadratic sequence: 5, 12, 23, 38, 57, ...

Solution: First difference: 7, 11, 15, 19 Second difference: 4, 4, 4 (Constant!) Finding a, b, and c: `2a = 4 => a = 2` `3a + b = 7 => 3(2) + b = 7 => b = 1` `a + b + c = 5 => 2 + 1 + c = 5 => c = 2` Therefore, `Tn = 2n² + n + 2`

Commentary: This question directly applies the method described above for finding the general term of a quadratic sequence. The key is to systematically calculate the first and second differences and then use the provided equations to solve for 'a', 'b', and 'c'.

Question 2: Calculate the sum of the first 15 terms of the arithmetic series: 2 + 5 + 8 + 11 + ...

Solution: Identify a and d: `a = 2` `d = 5 - 2 = 3` Apply the formula: `S15 = 15/2 [2(2) + (15-1)3]` `S15 = 7.5 [4 + (14)3]` `S15 = 7.5 [4 + 42]` `S15 = 7.5 [46]` `S15 = 345`

Commentary: This question reinforces the use of the arithmetic series sum formula. Correctly identifying 'a' and 'd' is crucial for obtaining the correct answer.

Question 3: The general term of a sequence is given by Tn = 3n² - 2n +

1. Determine the first three terms and show that it is a quadratic sequence.

Solution: Find T1, T2, and T3: `T1 = 3(1)² - 2(1) + 1 = 3 - 2 + 1 = 2` `T2 = 3(2)² - 2(2) + 1 = 12 - 4 + 1 = 9` `T3 = 3(3)² - 2(3) + 1 = 27 - 6 + 1 = 22` The sequence is 2, 9, 22,...

First difference: 9 - 2 = 7, 22 - 9 = 13 Second difference: 13 - 7 = 6 (Constant) Since the second difference is constant, the sequence is quadratic.

Commentary: This question tests the ability to work backwards, using a given general term to generate a sequence and then confirming its type.

Question 4: The sum of the first 'n' terms of an arithmetic series is given by Sn = 2n² + 3n. Determine the common difference of the series.

Solution: Find S1 and S2: `S1 = 2(1)² + 3(1) = 2 + 3 = 5` `S2 = 2(2)² + 3(2) = 8 + 6 = 14` Find T1 and T2: `T1 = S1 = 5` `T2 = S2 - S1 = 14 - 5 = 9` Find the common difference (d): `d = T2 - T1 = 9 - 5 = 4` Therefore, the common difference is

4. Commentary: This question requires careful application of the relationship between Sn and Tn. Finding S1 and S2 correctly is crucial. Independent Practice (Questions Only) Determine the general term of the quadratic sequence: 1, 7, 17, 31, 49, ...