GENERAL MATHEMATICS POWERPOINT FOR STRENGTHENED SHS

ARITHMETIC

1. Exploring Arithmetic and Geometric Sequences

2. What are Sequences? A sequence is an ordered list of numbers Each number in the sequence is called a term Sequences follow specific patterns Today we'll explore two important types: arithmetic and geometric

3. Arithmetic Sequences In an arithmetic sequence, the difference between each term is constant This constant difference is called the "common difference" Example: 2, 5, 8, 11, 14, ... What's the common difference in this sequence?

4. Geometric Sequences In a geometric sequence, each term is multiplied by a constant to get the next term This constant multiplier is called the "common ratio" Example: 3, 9, 27, 81, 243, ... What's the common ratio in this sequence?

5. Identifying Arithmetic Sequences To identify an arithmetic sequence: Calculate the difference between consecutive terms If the difference is constant, it's arithmetic Example: Is 4, 7, 10, 13, 16, ... arithmetic? What's the common difference?

6. Identifying Geometric Sequences To identify a geometric sequence: Divide each term by the previous term If the ratio is constant, it's geometric Example: Is 2, 6, 18, 54, 162, ... geometric? What's the common ratio?

7. General Term of Arithmetic Sequences The nth term of an arithmetic sequence is given by: aₙ = a₁ + (n - 1)d Where: aₙ is the nth term, a₁ is the first term, n is the position, d is the common difference How would you find the 20th term of 3, 7, 11, 15, ...?

8. General Term of Geometric Sequences The nth term of a geometric sequence is given by: aₙ = a₁ × r^(n-1) Where: aₙ is the nth term, a₁ is the first term, r is the common ratio, n is the position How would you find the 6th term of 2, 6, 18, 54, ...?

9. Arithmetic Sequence Example Consider the sequence: 5, 8, 11, 14, 17, ... Is it arithmetic? Why? What's the common difference? What's the 10th term? Try calculating it using the formula!

10. Geometric Sequence Example Consider the sequence: 3, 12, 48, 192, ... Is it geometric? Why? What's the common ratio? What's the 7th term? Use the formula to verify your answer!

11. Arithmetic Sequence Applications Arithmetic sequences appear in many real-world scenarios: Annual salary increases Linear distance traveled at constant speed Temperature changes at a steady rate Can you think of other examples?

12. Geometric Sequence Applications Geometric sequences are found in various applications: Compound interest Population growth Radioactive decay What other real-life situations might follow a geometric pattern?

13. Comparing Arithmetic and Geometric Growth Arithmetic growth is linear (constant increase) Geometric growth is exponential (multiplying by a constant) Example: Compare 2, 4, 6, 8, 10, ... (arithmetic) with 2, 4, 8, 16, 32, ... (geometric) Which grows faster? Why?

14. Arithmetic Sequence Sum Formula The sum of n terms in an arithmetic sequence is: Sₙ = (n/2)(a₁ + aₙ) Where: Sₙ is the sum, n is the number of terms, a₁ is the first term, aₙ is the last term How would you find the sum of the first 50 terms of 3, 7, 11, 15, ...?

15. Geometric Sequence Sum Formula The sum of n terms in a geometric sequence is: Sₙ = a₁(1 - r^n) / (1 - r) for r ≠ 1 Sₙ = na₁ for r = 1 Where: Sₙ is the sum, a₁ is the first term, r is the common ratio, n is the number of terms Calculate the sum of the first 6 terms of 2, 6, 18, 54, ...

16. Infinite Geometric Series Some geometric sequences have an infinite sum This occurs when |r| < 1 (the absolute value of r is less than 1) The formula for the sum of an infinite geometric series is: S∞ = a₁ / (1 - r) Can you think of a real-world example where this might apply?

17. Arithmetic Sequence Practice Given the arithmetic sequence 7, 13, 19, 25, ...: What is the common difference? Find the 15th term Calculate the sum of the first 20 terms Try these on your own, then we'll discuss!

18. Geometric Sequence Practice Given the geometric sequence 5, 15, 45, 135, ...: What is the common ratio? Find the 8th term Calculate the sum of the first 10 terms Work on these problems independently, then we'll review together

19. Sequences in Nature Many natural phenomena follow sequence patterns: Fibonacci sequence in flower petals and pinecones Logarithmic spirals in shells and galaxies Fractals in ferns and snowflakes How might understanding sequences help us study nature?

20. Review and Reflection We've explored arithmetic and geometric sequences Key differences: constant addition vs. constant multiplication Applications in finance, science, and nature What was the most interesting thing you learned? How might you use this knowledge in the future?