Sequence Formulas

Understanding sequence formulas is key in Honors Algebra II, Precalculus. These formulas help us analyze patterns in numbers, whether through consistent differences in arithmetic sequences or constant ratios in geometric sequences, laying the groundwork for more complex mathematical concepts.

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 constant 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 is given by ( 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 derived from the sum of the two preceding terms.

7. Explicit Sequence Formula

   • An explicit formula allows for direct computation of the nth term without needing previous terms.
   • It is typically expressed in the form ( a_n = f(n) ), where ( f(n) ) is a function of n.
   • This formula simplifies finding specific terms in a sequence without recursion.

8. 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 explicit formula, known as Binet's formula, is ( F_n = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}} ), where ( \phi ) is the golden ratio.
   • This sequence appears in various natural phenomena and is a classic example of recursive sequences.

9. Arithmetic-Geometric Sequence Formula

   • An arithmetic-geometric sequence combines elements of both arithmetic and geometric sequences.
   • The nth term can be expressed as ( a_n = (a_1 + (n-1)d) \cdot r^{(n-1)} ).
   • This sequence is useful in various applications, including financial calculations and certain mathematical models.

10. Binomial Theorem for Sequences

    • The Binomial Theorem provides a way to expand expressions of the form ( (a + b)^n ).
    • It states that ( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k ), where ( \binom{n}{k} ) is a binomial coefficient.
    • This theorem is fundamental in combinatorics and helps in understanding the distribution of terms in polynomial expansions.