general mathematics for strengthened shs .pptx

exploring geometric series

1. Exploring Arithmetic and Geometric Series

2. What are Sequences? A sequence is an ordered list of numbers Each number in the sequence is called a term Sequences follow a specific pattern Can you think of any number patterns you've seen before?

3. Introduction to Series A series is the sum of all terms in a sequence We use the Greek letter Σ (sigma) to represent summation Two common types of series: 1. Arithmetic series 2. Geometric series How do you think these two types might differ?

4. Arithmetic Sequences In an arithmetic sequence, the difference between terms is constant Example: 2, 5, 8, 11, 14, ... The common difference (d) here is 3 Can you identify the next two terms in this sequence?

5. Arithmetic Sequence Formula General term formula: aₙ = a₁ + (n - 1)d Where: aₙ = nth term a₁ = first term n = term number d = common difference How would you find the 10th term of the sequence 3, 7, 11, 15, ...?

6. Arithmetic Series An arithmetic series is the sum of terms in an arithmetic sequence Example: 2 + 5 + 8 + 11 + 14 + ... We can use a formula to find the sum quickly Why do you think finding a quick way to sum series is useful?

7. Arithmetic Series Formula Sum formula: Sₙ = n(a₁ + aₙ)/2 Where: Sₙ = sum of n terms n = number of terms a₁ = first term aₙ = last term This formula is based on taking the average of the first and last terms How would this help in calculating large sums quickly?

8. Arithmetic Series Example Find the sum of the first 50 positive integers First term (a₁) = 1, Last term (a₅₀) = 50 Using the formula: S₅₀ = 50(1 + 50)/2 = 1275 Can you think of a real-life application for this calculation?

9. Geometric Sequences In a geometric sequence, each term is a constant multiple of the previous term Example: 2, 6, 18, 54, 162, ... The common ratio (r) here is 3 What's the difference between this and an arithmetic sequence?

10. Geometric Sequence Formula General term formula: aₙ = a₁ * r^(n-1) Where: aₙ = nth term a₁ = first term r = common ratio n = term number How would you find the 7th term of the sequence 1, 3, 9, 27, ...?

11. Geometric Series A geometric series is the sum of terms in a geometric sequence Example: 2 + 6 + 18 + 54 + 162 + ... Like arithmetic series, we have formulas for quick calculations Why might geometric series be important in fields like finance?

12. Finite Geometric Series Formula Sum formula: Sₙ = a₁(1 - r^n)/(1 - r), where r ≠ 1 Where: Sₙ = sum of n terms a₁ = first term r = common ratio n = number of terms This formula efficiently calculates sums for large n How does this differ from the arithmetic series formula?

13. Infinite Geometric Series Some geometric series can be infinite For |r| < 1, we can find the sum of infinite terms Sum formula: S∞ = a₁/(1 - r), where |r| < 1 Why do you think this only works when |r| < 1?

14. Geometric Series Example Find the sum of 2 + 6 + 18 + 54 + ... to 6 terms First term (a₁) = 2, Common ratio (r) = 3, n = 6 Using the formula: S₆ = 2(1 - 3⁶)/(1 - 3) = 728 Can you verify this by adding the terms manually?

15. Real-World Application: Compound Interest Compound interest follows a geometric sequence Initial amount grows by a fixed percentage each period Formula: A = P(1 + r)^n Where: A = final amount, P = principal, r = interest rate, n = number of periods How does this relate to the geometric sequence formula?

16. Real-World Application: Population Growth Population growth can often be modeled by geometric sequences Example: A bacteria culture doubles every hour Starting with 100 bacteria, after 5 hours: 100 * 2⁵ = 3200 Can you think of other phenomena that might follow geometric growth?

17. Comparing Arithmetic and Geometric Series Arithmetic: Linear growth (constant addition) Geometric: Exponential growth (constant multiplication) Geometric series grow much faster for r > 1 Which type of growth would you prefer for your savings account? Why?

18. Series in Technology Computer algorithms often use series Example: Binary search divides search space in half each time (geometric) File compression can use geometric series principles How might understanding series help in developing efficient algorithms?

19. Puzzle Time! A ball bounces to 3/4 of its previous height each time it hits the ground If it starts at 10 meters, what's the total distance traveled after 5 bounces? Hint: This involves both arithmetic and geometric series! Try solving this in groups and discuss your approach

20. Review and Reflection We've explored arithmetic and geometric sequences and series Discussed their formulas and real-world applications What was the most surprising thing you learned? How might you use this knowledge in your future studies or career?

21. Questions and Further Exploration Any questions about arithmetic or geometric series? Interested in learning more? Consider exploring: Fibonacci sequence Taylor series Convergence of infinite series How could you apply what you've learned to your other subjects?