Pre-Cal 40S June 4, 2009

The Legacy of Karl

Fredrich Gauss that is ...

unstacking by flickr user mikelietzZehner by flickr user threedots

Allan is one of 7 men and Brigit is one of 10 women who wish to be chosen for the show The Greatest Mathematician. From this group, 4 men and 4 women will be chosen. What is the probability that both Allan and Brigit will be among the 8 people chosen? Briefly explain your calculations.

Sequences and Series on YouTube

Introduction to today's class by Mr. Green on YouTube ... a summary of almost everything in this unit ...

http://youtube.com/watch?v=WjLSz-nNLBc

Solution: a = 11 t51 = 11 + (51 - 1)(-6) d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289

Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ...

tn is the nth terma is the first termn is the "rank" of the nth term in the sequenced is the common difference

tn = a + (n - 1)d

To Find the nth Term In an Arithmetic Sequence

To Find the nth Term In a Geometic Sequence

r is the common ratio

n is the "rank" of the nth term in the sequence

a is the first term

tn is the nth term

Photo Source: Karl Gauss (1777–1855)

The Story of Young Gauss ...http://www.sigmaxi.org/amscionline/gauss-snippets.html

Artithmetic Series: The sum of numbers in an arithmetic sequence given by

Series: The sum of numbers in a sequence to a particular term in a sequence.

Example: denotes the sum of the first 5 terms. denotes the sum of the first n terms.

is the sum to the nth termn is the "rank" of the nth terma is the first term in the sequenced is the common difference

(a) What is the sum of the integers from 1 to 5000?

(b) What is the sum of all multiples of 7 between 1 & 5000?

(c) What is the sum of all integers from 1 to 5000 inclusive that are not multiples of 7?

Sigma Notation: A shorthand way to write a series.

(2n - 3) is the implicit definition of the sequence

superscript 4 means keep evaluating (2n - 3) for successive integral values of n; stop when n = 4; then add all the terms

subscript n = 1 means "start with n = 1 and evaluate (2n - 3)"

Σ is capital sigma (from the greek alphabet); means sum

means (2(1) -3) + (2(2) -3) + (2(3) -3) + (2(4) -3) = -1 + 1 + 3 + 5

= 8

∑n=1

4

(2n - 3)

Example:

Find the value of:

Series: The sum of numbers in a sequence to a particular term in a sequence.

Example: denotes the sum of the first 5 terms. denotes the sum of the first n terms.

Geometric Series: The sum of numbers in an geometric sequence given by

is the sum to the nth termn is the "rank" of the nth terma is the first term in the sequenced is the common difference

or