SUMMATION or SIGMA NOTATION

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Presentation transcript:

SUMMATION or SIGMA NOTATION Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

DEFINITIONS Sigma Notation for Finite Sums This symbol denotes the sum a1 + a2 + . . . + an. The a's are the terms of the sum: a1 is the first term, a2 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; the number n is the upper limit of summation. Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

Write the sum without sigma notation. Then evaluate it. EXAMPLE Write the sum without sigma notation. Then evaluate it. Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

SOLUTION Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

Write the sum without sigma notation. Then evaluate it. EXAMPLE Write the sum without sigma notation. Then evaluate it. Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

SOLUTION Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

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. Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

Express the sum 1 + 3 + 5 + 7 + 9 in sigma notation. EXAMPLE Express the sum 1 + 3 + 5 + 7 + 9 in sigma notation. Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

HERE ARE SEVERAL SOLUTIONS Starting with k = 2: Starting with k = -3: Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

...SOLUTION Starting with k = 0: Starting with k = 1: Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

ALGEBRA RULES FOR FINITE SUMS... Sum Rule: Difference Rule: Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

...ALGEBRA RULES FOR FINITE SUMS Constant Multiple Rule: Constant Value Rule: Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

Constant Multiple Rule EXAMPLE Sum Rule Constant Multiple Rule Constant Value Rule Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

Discovered by Gauss at age 5 The first n integers: FAMOUS FINITE SUM FORMULAS Discovered by Gauss at age 5 The first n squares: The first n cubes: Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

Use the algebra rules and the finite sum formulas to evaluate the sum. EXAMPLE Use the algebra rules and the finite sum formulas to evaluate the sum. Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy

Constant Multiple Rule SOLUTION Distribute the k. Sum Rule and Constant Multiple Rule Friday, February 15, 2019Friday, February 15, 2019 Mr M Kennedy Finite Sum Formulas