GENERAL MATHEMATICS POWERPOINT FOR STRENGTHENED SHS

SIGMA NOTATION

1. Mastering Sigma Notation: From Series to Summations and Back

2. Introduction to Series and Sigma Notation A series is a sum of terms in a sequence Sigma notation (Σ) is a compact way to represent series We'll learn how to convert between series and sigma notation Why is this important? It simplifies complex calculations!

3. What is a Series? A series is the sum of terms in a sequence Example: 1 + 2 + 3 + 4 + 5 Can you think of other examples of series in math or real life? Series can be finite (have an end) or infinite (go on forever)

4. Introduction to Sigma Notation Sigma notation uses the Greek letter Σ (sigma) It's a shorthand way to write sums General form: Σ(term) from i = start to end i is called the index of summation What do you think the advantages of using sigma notation might be?

5. Anatomy of Sigma Notation Σ: The summation symbol i: Index of summation Start value: Where the sum begins End value: Where the sum ends Term: The expression being summed Can you identify these parts in Σ(2i + 1) from i = 1 to 5?

6. Example: Converting a Series to Sigma Notation Series: 2 + 4 + 6 + 8 + 10 Identify the pattern: Each term is 2 times its position Write in sigma notation: Σ(2i) from i = 1 to 5 How would you explain this conversion to a classmate?

7. Practice: Series to Sigma Notation Convert these series to sigma notation: 1. 1 + 3 + 5 + 7 + 9 2. 1 + 4 + 9 + 16 + 25 3. 10 + 8 + 6 + 4 + 2 Hint: Look for patterns in the terms

8. Evaluating Sums Using Sigma Notation To evaluate, replace i with each value from start to end Example: Σ(2i + 1) from i = 1 to 4 = (2(1) + 1) + (2(2) + 1) + (2(3) + 1) + (2(4) + 1) = 3 + 5 + 7 + 9 = 24 Why is it important to be able to evaluate sums?

9. Practice: Evaluating Sums Evaluate these sums: 1. Σ(i) from i = 1 to 5 2. Σ(2i - 1) from i = 1 to 4 3. Σ(i²) from i = 1 to 3 What strategies did you use to solve these?

10. Converting Sigma Notation to a Series Replace i with each value from start to end Write out each term Example: Σ(3i - 2) from i = 1 to 4 = (3(1) - 2) + (3(2) - 2) + (3(3) - 2) + (3(4) - 2) = 1 + 4 + 7 + 10 How does this process compare to evaluating sums?

11. Practice: Sigma Notation to Series Convert these to series: 1. Σ(i + 3) from i = 0 to 3 2. Σ(2ⁱ) from i = 1 to 4 3. Σ(10 - i) from i = 1 to 5 What patterns do you notice in the resulting series?

12. Properties of Sigma Notation Constant multiple: Σ(k × aᵢ) = k × Σ(aᵢ) Sum of sums: Σ(aᵢ + bᵢ) = Σ(aᵢ) + Σ(bᵢ) Constant sum: Σ(c) = n × c, where n is number of terms How might these properties simplify calculations?

13. Using Sigma Notation Properties Example: Simplify Σ(3i + 2) from i = 1 to 4 Σ(3i + 2) = Σ(3i) + Σ(2) = 3Σ(i) + 4(2) = 3(1 + 2 + 3 + 4) + 8 = 3(10) + 8 = 38 Can you think of a real-life scenario where this might be useful?

14. Practice: Applying Sigma Notation Properties Simplify these expressions: 1. Σ(2i - 3) from i = 1 to 5 2. Σ(4i + 1) from i = 0 to 3 3. Σ(i² + 2i) from i = 1 to 3 Which properties did you use for each problem?

15. Changing the Index of Summation Sometimes it's useful to shift the index Example: Σ(i + 1) from i = 0 to 3 Can be rewritten as: Σ(i) from i = 1 to 4 Both sum 1 + 2 + 3 + 4 Why might changing the index be helpful?

16. Practice: Changing the Index Rewrite these sums with a different index: 1. Σ(2i + 1) from i = 1 to 5 2. Σ(i²) from i = 0 to 4 3. Σ(1/i) from i = 1 to n How does changing the index affect the terms and limits?

17. Infinite Series and Sigma Notation Infinite series continue without end Represented in sigma notation with ∞ as the upper limit Example: Σ(1/2ⁱ) from i = 0 to ∞ Some infinite series converge, others diverge Can you think of a real-world example of an infinite process?

18. Applications of Series and Sigma Notation Physics: Calculating total force or energy Finance: Compound interest calculations Computer Science: Analyzing algorithm efficiency Statistics: Calculating expected values Where else might you encounter series in your studies or daily life?

19. Common Series and Their Sigma Notation

20. Review and Key Takeaways Series are sums of terms in a sequence Sigma notation compactly represents series Converting between series and sigma notation is a key skill Properties of sigma notation simplify calculations Series and sigma notation have wide-ranging applications What was the most surprising thing you learned about sigma notation?

21. Final Challenge: Create Your Own Series Create a series with at least 5 terms Write it in expanded form Represent it using sigma notation Explain the pattern to a classmate How does creating your own series deepen your understanding?