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Key Concepts

12.1 Sequences

  • Factorial Notation
    If n is a positive integer, then n!n! is
    n!=n(n1)(n2)(3)(2)(1).n!=n(n1)(n2)(3)(2)(1).

    We define 0!0! as 1, so 0!=10!=1
  • Summation Notation
    The sum of the first n terms of a sequence whose nth term anan is written in summation notation as:
    i=1nai=a1+a2+a3+a4+a5++ani=1nai=a1+a2+a3+a4+a5++an

    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 a1a1 and the common difference d is
    an=a1+(n1)dan=a1+(n1)d
  • Sum of the First n Terms of an Arithmetic Sequence
    The sum, Sn,Sn, of the first n terms of an arithmetic sequence, where a1a1 is the first term and anan is the nth term is
    Sn=n2(a1+an)Sn=n2(a1+an)

12.3 Geometric Sequences and Series

  • General Term (nth term) of a Geometric Sequence: The general term of a geometric sequence with first term a1a1 and the common ratio r is
    an=a1rn1an=a1rn1
  • Sum of the First n Terms of a Geometric Series: The sum, Sn,Sn, of the n terms of a geometric sequence is
    Sn=a1(1rn)1rSn=a1(1rn)1r

    where a1a1 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 a1a1 and common ratio is r and is written
    a1+a1r+a1r2++a1rn1+a1+a1r+a1r2++a1rn1+
  • Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is a1a1 and common ratio r,
    If|r|<1,the sum is S=a11r We say the series converges. If|r|1,the infinite geometric series does not have a sum. We say the series diverges.If|r|<1,the sum is S=a11r We say the series converges. If|r|1,the infinite geometric series does not have a sum. We say the series diverges.
  • Value of an Annuity with Interest Compounded nn 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
    At=P((1+rn)nt1)rnAt=P((1+rn)nt1)rn

12.4 Binomial Theorem

  • Patterns in the expansion of (a+b)n(a+b)n
    • The number of terms is n+1.n+1.
    • The first term is anan and the last term is bn.bn.
    • 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 This figure shows Pascal’s Triangle. The first level is 1. The second level is 1, 1. The third level is 1, 2, 1. The fourth level is 1, 3, 3, 1. The fifth level is 1, 4, 6, 4, 1. The sixth level is 1, 5, 10, 10, 5, 1. The seventh level is 1, 6, 15, 20, 15, 6, 1
  • Binomial Coefficient (nr)(nr) : A binomial coefficient (nr),(nr), where r and n are integers with 0rn,0rn, is defined as
    (nr)=n!r!(nr)!(nr)=n!r!(nr)!

    We read (nr)(nr) as “n choose r” or “n taken r at a time”.
  • Properties of Binomial Coefficients
    (n1)=n(nn)=1(n0)=1 (n1)=n(nn)=1(n0)=1
  • Binomial Theorem: For any real numbers a, b, and positive integer n,
    (a+b)n=(n0)an+(n1)an1b1+(n2)an2b2+...+(nr)anrbr+...+(nn)bn(a+b)n=(n0)an+(n1)an1b1+(n2)an2b2+...+(nr)anrbr+...+(nn)bn
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