SUMMATION or SIGMA NOTATION 372021 Mr M Kennedy

SUMMATION or SIGMA NOTATION 3/7/2021 Mr M Kennedy 1

DEFINITIONS Sigma Notation for Finite Sums • This symbol denotes the sum a 1 + a 2 +. . . + an. • The a's are the terms of the sum: a 1 is the first term, a 2 is the second term, ak is the kth term, and an is the nth and last term. • The variable k is the index of summation. • The values of k run through the integers from 1 to n. • The number 1 is the lower limit of summation; 3/7/2021 Mr M Kennedy 2 the number n is the upper limit of summation.

EXAMPLE Write the sum without sigma notation. Then evaluate it. 3/7/2021 Mr M Kennedy 3

SOLUTION 3/7/2021 Mr M Kennedy 4

EXAMPLE Write the sum without sigma notation. Then evaluate it. 3/7/2021 Mr M Kennedy 5

SOLUTION 3/7/2021 Mr M Kennedy 6

NOTE The lower limit of summation does not have to be 1; it can be any integer. It is often simplest to start with k = 0 or k = 1. 3/7/2021 Mr M Kennedy 7

EXAMPLE Express the sum 1+3+5+7+9 in sigma notation. 3/7/2021 Mr M Kennedy 8

HERE ARE SEVERAL SOLUTIONS Starting with k = 2: Starting with k = -3: 3/7/2021 Mr M Kennedy 9

. . . SOLUTION Starting with k = 0: Starting with k = 1: 3/7/2021 Mr M Kennedy 10

ALGEBRA RULES FOR FINITE SUMS. . . Sum Rule: Difference Rule: 3/7/2021 Mr M Kennedy 11

. . . ALGEBRA RULES FOR FINITE SUMS Constant Multiple Rule: Constant Value Rule: 3/7/2021 Mr M Kennedy 12

EXAMPLE Sum Rule Constant Multiple Rule Constant Value Rule 3/7/2021 Mr M Kennedy 13

The first n FAMOUS FINITE SUM integers: integers FORMULAS Discovered by Gauss at age 5 The first n squares: squares The first n cubes: cubes 3/7/2021 Mr M Kennedy 14

EXAMPLE Use the algebra rules and the finite sum formulas to evaluate the sum. 3/7/2021 Mr M Kennedy 15

SOLUTION Distribute the k. Sum Rule and Constant Multiple Rule 3/7/2021 Mr M Kennedy Finite Sum Formulas 16