Lesson Notes By Weeks and Term v5 - Grade 12

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

• Solve problems with arithmetic and geometric sequences/series.
• Apply the formula for the sum to infinity of a convergent series.
• Identify convergent and divergent geometric series.
• Use and manipulate sigma notation.
• Apply sequences and series to solve real-world problems.

Lesson summary

This week, we delve deeper into patterns, sequences, and series, building upon the foundational knowledge acquired in Grade 11 and the previous week's lesson. Understanding sequences and series is not just about manipulating numbers; it's about recognizing patterns in the world around us and using them to make predictions and solve problems. From calculating loan repayments to predicting population growth, sequences and series provide powerful mathematical tools.

Lesson notes

2.1 Arithmetic Sequences and Series An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by d.

General Term (nth term): T n = a + (n - 1)d Where: T n is the nth term of the sequence a is the first term n is the term number d is the common difference Sum of the first n terms: S n = n/2 [2a + (n - 1)d] or S n = n/2 (a + L), where L is the last term (T n ).

Example 1: Consider the arithmetic sequence: 2, 5, 8, 11, ... a = 2 (first term) d = 3 (common difference)

Let's find the 10th term (T 10 ): T 10 = 2 + (10 - 1) 3 = 2 + 9 3 = 2 + 27 = 29 Now let's find the sum of the first 10 terms (S 10 ): S 10 = 10/2 [2(2) + (10 - 1) 3] = 5 [4 + 27] = 5 31 = 155 2.2 Geometric Sequences and Series A geometric sequence is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio, denoted by r.

General Term (nth term): T n = a r (n - 1)

Where: T n is the nth term of the sequence a is the first term n is the term number r is the common ratio Sum of the first n terms: S n = a(r n - 1) / (r - 1), if r ≠ 1 Example 2: Consider the geometric sequence: 3, 6, 12, 24, ... a = 3 (first term) r = 2 (common ratio)

Let's find the 7th term (T 7 ): T 7 = 3 2 (7 - 1) = 3 2 6 = 3 * 64 = 192 Now let's find the sum of the first 7 terms (S 7 ): S 7 = 3(2 7 - 1) / (2 - 1) = 3(128 - 1) / 1 = 3 * 127 = 381 2.3 Convergent and Divergent Geometric Series A geometric series converges if the sum of its terms approaches a finite limit as the number of terms approaches infinity. This happens when |r| ∞ ):** S ∞ = a / (1 - r), if |r| ∞ = 1 / (1 - 1/2) = 1 / (1/2) = 2 Example 4: Consider the geometric series: 2 + 4 + 8 + 16 + ... a = 2 (first term) r = 2 (common ratio) Since |r| = |2| > 1, the series diverges. It does not have a finite sum to infinity. 2.4 Sigma Notation Sigma notation (Σ) is a shorthand way of representing a series. It indicates the sum of a sequence of terms.

The general form is: ∑ k=m n f(k)

Where: Σ (sigma) indicates summation k is the index of summation m is the lower limit of summation (starting value of k) n is the upper limit of summation (ending value of k) f(k) is the expression being summed Example 5: ∑ k=1 5 k 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 = 1 + 4 + 9 + 16 + 25 = 55 Example 6: Write the following series using sigma notation: 4 + 7 + 10 + 13 + 16 This is an arithmetic series with a = 4 and d =

3. The general term is T n = 4 + (n-1)3 = 3n +

1. We have 5 terms. So, the series can be written as: ∑ n=1 5 (3n + 1) Guided Practice (With Solutions)

Question 1: Find the sum of the first 8 terms of the arithmetic sequence where the first term is -3 and the common difference is

4. Solution: a = -3, d = 4, n = 8 S n = n/2 [2a + (n - 1)d] S 8 = 8/2 [2(-3) + (8 - 1)4] = 4 [-6 + 28] = 4 22 = 88

Commentary: This question directly applies the formula for the sum of an arithmetic series. It's important to correctly substitute the given values into the formula.

Question 2: Determine the sum to infinity of the geometric series: 9 + 3 + 1 + 1/3 + ...

Solution: a = 9, r = 3/9 = 1/3 Since |r| = |1/3| ∞ = a / (1 - r) = 9 / (1 - 1/3) = 9 / (2/3) = 9 (3/2) = 27/2 = 13.5

Commentary: This question tests the understanding of convergent geometric series and the application of the sum to infinity formula. Remember to check that |r| k=2 6 (2k - 1)

Solution: ∑ k=2 6 (2k - 1) = (2(2) - 1) + (2(3) - 1) + (2(4) - 1) + (2(5) - 1) + (2(6) - 1) = (4 - 1) + (6 - 1) + (8 - 1) + (10 - 1) + (12 - 1) = 3 + 5 + 7 + 9 + 11 = 35

Commentary: This question tests the ability to expand a series expressed in sigma notation. Carefully substitute each value of k within the specified range.

Question 4: In an arithmetic sequence, the 3rd term is 7 and the 7th term is

1

5. Determine the first term and the common difference.

Solution: T 3 = a + 2d = 7 T 7 = a + 6d = 15 Subtracting the first equation from the second, we get: 4d = 8, so d = 2 Substituting d = 2 into the first equation: a + 2(2) = 7 a + 4 = 7 a = 3 Therefore, a = 3 and d =

2. Commentary: This questions requires solving simultaneous equations. We set up two equations based on the information about the 3rd and 7th terms and then solve for 'a' and 'd'. Independent Practice (Questions Only) Find the 15th term of the arithmetic sequence: 4, 7, 10, 13, ... Determine the sum of the first 20 terms of the arithmetic series: -5 - 2 + 1 + 4 + ... The first term of a geometric sequence is 5 and the common ratio is

2. Find the sum of the first 6 terms. Determine the sum to infinity of the geometric series: 16 + 8 + 4 + 2 + ...

Evaluate the following expression: ∑ k=1 4 (k 2 + 1) Write the arithmetic series 2 + 6 + 10 + 14 + 18 using sigma notation. The 5th term of an arithmetic sequence is 23 and the 12th term is

5

1. Find the first term and the common difference. A geometric series has a first term of 3 and a common ratio of -1/

2. Find the sum to infinity.