Intermezzo

Listen to the music ... the geometric sequences music ....

Evan Ma - Sequences and Series

http://youtube.com/watch?v=M1WzkWy3n74

Photo Source: Karl Gauss (1777–1855)

The Story of Young Gauss ...

to be continued ...

SEQUENCE

SERIES

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.

Artithmetic Series: The sum of numbers in an arithmetic 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

(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.

Example: means (2(1) -3) + (2(2) -3) + (2(3) -3) + (2(4) -3) = -1 + 1 + 3 + 5 = 8

Σ is capital sigma (from the greek alphabet); means sumsubscript n = 1 means "start with n = 1 and evaluate (2n - 3)"superscript 4 means keep evaluating (2n - 3) for successive integral

values of n; stop when n = 4; then add all the terms(2n - 3) is the implicit definition of the sequence

Find the value of:

Given a geometric sequence in which and , what is the value of ?