Sequences and Summations

Sequences

Definition of a sequence A sequence is a function from a subset of integers to a set S.

An example of a sequence Beginning with n=1

A geometric progression Beginning with n=0

An arithmetic progression Beginning with n=0

Example progression

Summations

Summations Sigma for “sum”

Product Pi for “product”

Summations Upper limit Lower limit Index of summation

Summations example The sum of the first 100 integers What’s the answer?

Summations example/picture We now have a square 10 (n) by 11 (n+1) with area 110 units We need half of that (10x11)/2

Summations challenge

Summations challenge

Summations: shifting indices

Summations: a theorem Proof: see the book

Summations: nesting

Summations: implementing a nesting as a bit of program for i = 1 to n-1 do for j=i+1 to n do sum = sum + i*j

Summations: just a wee tweak

One More Example

One More Example Table 2, page 157

One More Example Then use this

Challenge