Sequences Formulas

Understanding sequences and their formulas is key in Honors Pre-Calculus. These concepts, like arithmetic and geometric sequences, help us analyze patterns and sums, laying the groundwork for more complex mathematical ideas and real-world applications.

1. Arithmetic Sequence Formula

   • An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.
   • The formula for the nth term is given by ( a_n = a_1 + (n-1)d ), where ( a_1 ) is the first term and ( d ) is the common difference.
   • The sequence can be represented as ( a_1, a_1 + d, a_1 + 2d, \ldots ).

2. Geometric Sequence Formula

   • A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
   • The formula for the nth term is ( a_n = a_1 \cdot r^{(n-1)} ), where ( a_1 ) is the first term and ( r ) is the common ratio.
   • The sequence can be expressed as ( a_1, a_1 \cdot r, a_1 \cdot r^2, \ldots ).

3. Arithmetic Series Sum Formula

   • An arithmetic series is the sum of the terms of an arithmetic sequence.
   • The sum of the first n terms can be calculated using ( S_n = \frac{n}{2} (a_1 + a_n) ) or ( S_n = \frac{n}{2} (2a_1 + (n-1)d) ).
   • This formula allows for quick calculation of the total when the number of terms and the first and last terms are known.

4. Geometric Series Sum Formula

   • A geometric series is the sum of the terms of a geometric sequence.
   • The sum of the first n terms is given by ( S_n = a_1 \frac{1 - r^n}{1 - r} ) (for ( r \neq 1 )).
   • This formula is useful for calculating the total of a finite geometric series.

5. Infinite Geometric Series Sum Formula

   • An infinite geometric series is the sum of an infinite number of terms in a geometric sequence.
   • The sum converges to ( S = \frac{a_1}{1 - r} ) if the absolute value of the common ratio ( |r| < 1 ).
   • This formula is essential for understanding limits and convergence in sequences.

6. Recursive Sequence Formula

   • A recursive sequence defines each term based on one or more previous terms.
   • The general form is ( a_n = f(a_{n-1}, a_{n-2}, \ldots) ) with initial conditions specified.
   • This approach is useful for sequences like the Fibonacci sequence, where each term is the sum of the two preceding terms.

7. Explicit Sequence Formula

   • An explicit formula provides a direct way to calculate the nth term of a sequence without needing previous terms.
   • It is often expressed in the form ( a_n = f(n) ), where f is a function of n.
   • This formula simplifies finding specific terms in a sequence.

8. Sigma Notation

   • Sigma notation is a concise way to represent the sum of a sequence of terms.
   • It is written as ( \sum_{i=m}^{n} f(i) ), where m is the starting index, n is the ending index, and f(i) is the function defining the terms.
   • This notation is crucial for working with series and understanding summation in calculus.

9. Arithmetic-Geometric Sequence Formula

   • An arithmetic-geometric sequence combines elements of both arithmetic and geometric sequences.
   • Each term is formed by multiplying an arithmetic term by a geometric term, typically expressed as ( a_n = (a_1 + (n-1)d) \cdot r^{(n-1)} ).
   • This sequence is useful in various applications, including financial calculations and algorithm analysis.

10. Fibonacci Sequence Formula

    • The Fibonacci sequence is defined recursively as ( F_n = F_{n-1} + F_{n-2} ) with initial conditions ( F_0 = 0 ) and ( F_1 = 1 ).
    • The sequence starts as 0, 1, 1, 2, 3, 5, 8, and so on.
    • It appears in various natural phenomena and has applications in computer science, art, and nature.