- Factorial Notation
If n is a positive integer, then is
We define as 1, so
- Summation Notation
The sum of the first n terms of a sequence whose nth term is written in summation notation as:
The i is the index of summation and the 1 tells us where to start and the n tells us where to end.
12.2 Arithmetic Sequences
- General Term (nth term) of an Arithmetic Sequence
The general term of an arithmetic sequence with first term and the common difference d is
- Sum of the First n Terms of an Arithmetic Sequence
The sum, of the first n terms of an arithmetic sequence, where is the first term and is the nth term is
12.3 Geometric Sequences and Series
- General Term (nth term) of a Geometric Sequence: The general term of a geometric sequence with first term and the common ratio r is
- Sum of the First n Terms of a Geometric Series: The sum, of the n terms of a geometric sequence is
where is the first term and r is the common ratio.
- Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is and common ratio is r and is written
- Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is and common ratio r,
- Value of an Annuity with Interest Compounded Times a Year: For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is
12.4 Binomial Theorem
- Patterns in the expansion of
- The number of terms is
- The first term is and the last term is
- The exponents on a decrease by one on each term going left to right.
- The exponents on b increase by one on each term going left to right.
- The sum of the exponents on any term is n.
- Pascal’s Triangle
- Binomial Coefficient
: A binomial coefficient where r and n are integers with is defined as
We read as “n choose r” or “n taken r at a time”.
- Properties of Binomial Coefficients
- Binomial Theorem: For any real numbers a, b, and positive integer n,