Pre Calc Notes

Sums Using Sigma Notation

  • The discussion covers calculation of sums using sigma notation.

Sigma Notation Definition

  • The general formula for the sum of a series:
      - Sn=∑n=1N(a+(n−1)d)S_n = \sum_{n=1}^{N} (a + (n-1)d); where:
        - ( a ) = first term
        - ( d ) = common difference
        - ( n = 1, 2, …, N )

  • Sn representation:
      - For example, given: Sn=∑n=110(1+(n−1)⋅2)S_n = \sum_{n=1}^{10} (1 + (n-1)\cdot2)
        - Application for n = 1 to n = 10:
          - Calculating the sum gives:
            - When substituting back into the formula, it results in a total of 100 after simplification.

Infinite Geometric Series

  • The formula for an infinite geometric series:
      - S=a1−rS = \frac{a}{1 - r}; where:
        - ( a ) = first term
        - ( r ) = common ratio

  • Example Calculation:
      - If the first term is 256 and the common ratio is ( \frac{1}{2} ), then:
        - The series converges and results in a final answer based on common ratio calculations.

Types of Series

Arithmetic Series

  • The formula for the nth term of an arithmetic series is:
      - an=a+(n−1)⋅da_n = a + (n-1)\cdot d; where:
        - ( d ) = common difference

  • Calculation example:
      - For a series:
      - Given terms like 3, 7, …, compute the sum.
        - Executor a series where n is defined for specific terms.

Geometric Series

  • nth term representation for a geometric series is:
      - an=a⋅rn−1a_n = a\cdot r^{n-1}; where:
        - ( r ) = common ratio

  • Example working with a geometric sequence involves first term values supplied and how they evolve with each subsequent term being multiplied.

Sequence Types

  • Define specific sequence types:

Arithmetic Sequence

  • The general nth term formula for an arithmetic sequence:
      - Example: an=a+(n−1)⋅da_n = a + (n-1)\cdot d

  • Example values provided in sequences varying from 39 to negative results as part of arithmetic sequences.

Geometric Sequence

  • nth term formula for the geometric sequence:
      - Example: an=a⋅rn−1a_n = a\cdot r^{n-1}

  • Values highlighted in a typical sequence going negative or exponentially large based on the ratio used.