Merced College; Don Power

 

INTERMEDIATE ALGEBRA - CH 11, LECTURE

 

Sequences, Series and the Binomial Theorem

 

McKeague:

 

8.1  Sequences

 

Def of a sequence:  a function whose domain is the set of positive integers {1, 2, 3, ...}

Ex:  an = {11-2n} gives the sequence 9, 7, 5, 3, 1, -1, -3, ...

            Compare with the continuous function f(x) = 11-2x, with domain restricted to {1, 2, 3, ...}:

                        function values are f(1)=9, f(2)=7, etc.

            The difference is that the sequence would graph as discrete, unconnected points.

Notation:  Use "n" to indicate the number in the domain (independent variable).

            Use a1, a2, a3, etc. to indicate the function values (instead of f(1), f(2), f(3), etc.

            Use an = 11-2n instead of f(x) = 11-2x to indicate the general formula for the seqence

                        (called the general term or "nth term")

 

Examples:

            Find the first four terms of a sequence, given the general term

                        text example an = 2n-1 (an increasing sequence)

                        text example an = 1 / (n+1)  (a decreasing sequence)

                        text example an = (-1)n / n2  (an alternating sequence, because signs alternate)

 

Recursion formulas - sequence is defined by (1) identifying the first term(s), and then (2) defining each term in terms of the previous term(s) [rather than basing the formula on n]

            Ex:  (same as original sequence):  a1 = 9, an = an-1-2

 

Finding the general term of a sequence (tougher):

            Try showing both n and an in a table:  What do you have to do to n to get an?

            Consider looking at differences from one term to the next; if constant "c" compare cn with an

                        Ex:  5, 8, 11, 14, ...      (what would the previous term be?)

            Consider looking at ratios from one term to the next;  if constant "r" compare rn with an 

                        Ex:  20, 10, 5, 5/2, ...   (what would the previous term be?)

            Consider looking at powers of n (e.g. n2, n3, ...)

                        Ex:  -1, 2, 7, 14, ...      (what is the difference between each term and n2?

 

8.2  Series      

 

Def:  the sum of a number of terms in a sequence is a series.

            Finite

            Infinite

Summation notation (sigma notation)

            How to read :  the sum of the sequence 3i-1 from i=1 to i=5
                        i.e. 3(1)-1 + 3(2)-1 + 3(3)-1 + 3(4)-1 + 3(5)-1  =  2+5+8+11+14 = 40

            Terminology:

                        Sigma (capital Greek letter for S) for "sum"

                        index [of summation] -- here, the index is "i"  -- this is the [only] variable

                        summand - the expression being added

                        lower and upper limits (here, 1 and 5):  give the first and last values to be used for i.

 

Task:  Expand a summation

            Ex 1 - Like our example above

            Ex 2 - The index can be an exponent, ...

            Ex 3 - Other letters (besides the index) are treated as though they are constants -- they get copied.

Task:  Write a sum such as 1+3+5+7+9 with summation notation

            First, find a general term for the sequence

            Then turn it into a summation:  what are the first and last values of the index?

 

8.3  Arithmetic Sequences

 

Arithmetic Sequences (mentioned in R-8)

            Def:  Seq formed by adding the same value ("common difference") to each term

            Ex:  -3, 1, 5, ...

            Finding the common difference:  subt successive terms (later minus earlier)

General term of an "arithmetic progression" with first term a1 and common difference d is
            an = a1 + (n-1)d

Sum of the first n terms of an "arithmetic progression" with first term a1 and last term an is
            Sn = n/2 * (a1 + an)  =  nr of terms "n" times avg of terms (sum of first and last, over 2).

Called a "partial sum" because it sums only a finite part of the series

 

8.4  Geometric Sequences

 

Geometric Sequences (mentioned in R-8)

            Def:  Seq formed by multiplying the same value ("common ratio") times each term

            Ex:  10, 5, 5/2, 5/4, ...

            Finding the common ratio:  divide successive terms (later / earlier)

General term of a "geometric progression" with first term a1 and common ratio r is
            an = a1 r n-1

Partial sum of the first n terms of a "geometric progression" is
            Sn = a1 * (r n - 1) / (r  - 1)

            Proof:  Write out expressions for Sn and r*Sn; subtract and solve for Sn

                        Note that the last exponent in expansion of  Sn is n-1 (because exp. of 1st term is 0)

Infinite sum:  (valid only if |r| < 1)

            S = a1 / (1-r)                Proof:  let n increase (in previous formula) without bound.

 

8.5  The Binomial Expansion

 

You can get coefficients from Pascal's triangle

            How to number rows and columns.

Binomial coefficient formula uses factorials:  "n choose r" = n! / [r! * (n-r)!]

Def of factorial

            Meaning of 1! and 0!

Simplifying the formula by expanding and reducing

 

Return to:  Merced College; Don Power               Updated 11/03/08 by Don Power