general mathematics for grade 11 strengthened shs

mastering arithmetic

1. Mastering Arithmetic and Geometric Series

2. Introduction to Sequences and Series A sequence is an ordered list of numbers A series is the sum of the terms in a sequence We'll focus on two types: arithmetic and geometric How many sequences can you spot in everyday life?

3. Arithmetic Sequences Each term differs from the previous by a constant amount This constant is called the "common difference" (d) Example: 2, 5, 8, 11, 14, ... What's the common difference in this sequence?

4. Arithmetic Series Sum of terms in an arithmetic sequence Formula: S_n = n(a₁ + aₙ)/2 Where n is number of terms, a₁ is first term, aₙ is last term Can you explain why this formula works?

5. Problem: Arithmetic Series The first term of an arithmetic series is 3 and the last term is 51 There are 17 terms in total What is the sum of all terms? Try solving this before we move on!

6. Solution: Arithmetic Series Use the formula: S_n = n(a₁ + aₙ)/2 S_17 = 17(3 + 51)/2 S_17 = 17 × 54/2 = 459 The sum of all terms is 459 Did you get the correct answer?

7. Geometric Sequences Each term is a constant multiple of the previous term This constant is called the "common ratio" (r) Example: 2, 6, 18, 54, 162, ... What's the common ratio in this sequence?

8. Geometric Series Sum of terms in a geometric sequence Formula: S_n = a(1-rⁿ)/(1-r) when r ≠ 1 Where a is first term, r is common ratio, n is number of terms Why do you think we need r ≠ 1?

9. Problem: Geometric Series A geometric series has 6 terms The first term is 5 and the common ratio is 2 What is the sum of all terms? Try to solve this on your own!

10. Solution: Geometric Series Use the formula: S_n = a(1-rⁿ)/(1-r) S_6 = 5(1-2⁶)/(1-2) S_6 = 5(1-64)/(-1) = 5 × 63 = 315 The sum of all terms is 315 How close was your answer?

11. Infinite Geometric Series When |r| < 1, the series converges as n approaches infinity Formula for sum: S_∞ = a/(1-r) Example: 1 + 1/2 + 1/4 + 1/8 + ... Can you find the sum of this infinite series?

12. Applications: Arithmetic Series Calculating total distance in uniform motion Finding the sum of consecutive integers Determining the number of objects in triangular arrangements Can you think of other real-world applications?

13. Applications: Geometric Series Compound interest calculations Population growth models Calculating the total area of fractal shapes Where have you encountered geometric growth in real life?

14. Problem: Real-world Arithmetic Series A theater has 20 rows of seats The first row has 15 seats, and each row after has 2 more seats How many total seats are in the theater? Hint: What kind of series is this?

15. Solution: Real-world Arithmetic Series This is an arithmetic series with 20 terms First term (a₁) = 15, Last term (a₂₀) = 15 + 19 × 2 = 53 Use S_n = n(a₁ + aₙ)/2 S_20 = 20(15 + 53)/2 = 680 There are 680 seats in total Did this problem remind you of any real-life situations?

16. Problem: Real-world Geometric Series You invest $1000 at 8% annual interest, compounded yearly How much money will you have after 5 years? Hint: Think about how your money grows each year

17. Solution: Real-world Geometric Series This forms a geometric sequence with r = 1.08 Use the formula: A = P(1 + r)ⁿ A = 1000(1.08)⁵ ≈ $1469.33 You'll have about $1469.33 after 5 years How does this compare to simple interest?

18. Arithmetic vs Geometric: When to Use Arithmetic: Linear growth, constant increase/decrease Geometric: Exponential growth/decay, constant ratio Examples: Salary increases (A) vs. Compound interest (G) Can you classify these: Population growth, Distance traveled at constant speed?

19. Common Mistakes to Avoid Confusing arithmetic and geometric sequences Forgetting to check if r = 1 in geometric series Misusing infinity symbol in finite series What other mistakes have you encountered?

20. Practice Makes Perfect Solve a variety of problems regularly Focus on real-world applications Explain your reasoning to others Remember: Every expert was once a beginner!

21. Conclusion and Next Steps We've covered basics of arithmetic and geometric series These concepts are crucial in advanced math and real-world applications Next: Explore more complex series and their applications What aspect of series intrigues you the most?