Geometric Sequences Notes

Introduction to Geometric Sequences

• In previous lessons, we learned about sequences with addition.
• This lesson focuses on sequences where a certain number is multiplied.

Key Concepts

• Geometric Sequence: A sequence where each term is found by multiplying the previous term by a constant called the common ratio (r).
• The common ratio can be found by dividing any term in the sequence by the term that precedes it.

Finding the Common Ratio

1. Example 1: Sequence: 2, 8
   • Ratio: 8 / 2 = 4
2. Example 2: Sequence: -3, 9
   • Ratio: 9 / -3 = -3
3. Example 3: Sequence: 1, 1/2
   • Ratio: 1/2 / 1 = 1/2

Identifying Common Ratios and Next Terms

1. Example 1: 1, 2, 4, 8
   • Common Ratio: 8 / 4 = 2
   • Next Term: 8 * 2 = 16
2. Example 2: 80, 20, 5
   • Common Ratio: 20 / 80 = 1/4
   • Next Term: 5 * 1/4 = 5/4
3. Example 3: 2, -8, 32, -128
   • Common Ratio: -8 / 2 = -4
   • Next Term: -128 * -4 = 512*

Determining if a Sequence is Geometric

• Check if all terms share a common ratio.

1. Example 1: 5, 20, 80
   • Ratios: 20/5 = 4, 80/20 = 4
   • Geometric Sequence confirmed.
2. Example 2: 7√2, 5√2, 3√2, √2
   • Ratios do not match, so not a geometric sequence.
3. Example 3: 5, -10, 20, -40
   • Common ratio is -2, so is a geometric sequence.
4. Example 4: 10/3, 10/6, 10/9, 10/15
   • Ratios do not match, so not a geometric sequence.

n-th Term of a Geometric Sequence

• Formula for the n-th term:
  [ a_n = a_1 imes r^{(n - 1)} ]
  where:
  • ( a_1 ) = first term
  • ( r ) = common ratio
  • ( n ) = number of terms

Example Calculation

• Find the 10th term of the sequence 8, 4, 2, 1:
  • Common ratio (r): 4 / 8 = 1/2
  • First term (a1): 8
  • Calculation:
    [ a_{10} = 8 imes (1/2)^{(10-1)} = 8 imes (1/2)^9 ]
    • Result: ( a_{10} = 1/64 )

Exercises

1. Find the missing term in the sequence 3, 12, 48:
   • Result: 192
2. Given sequence: blank, blank, 32, 64, 128:
   • Result: 8 and 16

Real-World Application Problem

• Problem: Number of infections during a measles outbreak grows geometrically: 4, 8, 16.
  • Question: How many will be infected on the sixth day?
  • Solution:
    [ a_6 = 4 imes 2^{(6-1)} = 4 imes 32 = 128 ]
  • Result: 128 people will be infected.

Conclusion

• Geometric sequences can be identified by their common ratio and allow for calculations of future terms using the n-th term formula.