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23 Summation Notation
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Chapters and Appendices

Book Entrance

Foreword
1. Introduction
2. Implication Rules [4]
3. Chains of Reason [3]
4. Induction Mathematical
4. Romeo and Juliet
6  Old Language
5 Knowledge Islands [2]
7  Arith Skill Check [4 X 2]
Arith Webvideos
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable [8]
9. Algebra Talk [7]
10 Two More Skills[5]
11 Why Shorthand
12 Shorthand Usage [10]
13 What's Next
PS: The 4-th Skill For Algebra
14 Compound Interest [6]
15 Linear Equations [5]
16 Painless Proofs
17 Pythagoras
PS I.  Distributive Law
PS II. Polynomials
18 Rules of Algebra [20]
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums [2]
23 Summation Notation
24 Your Money [3]
25 Induction & Recursion [4]
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
Pathways for Learning

Book Entrance

Would you like to show yourself or others how to be  algebra power users?

What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

Chapter 23
Notation For Sums

1  Summation Notation

Mathematical shorthand for the sum

S = f(3)+f(4)+f(5)+f(6)

is S = å_{j = 3}⁶f(j) or

S = 6 å j = 3  f(j)

Here å is the Greek letter for capital sigma. The small or lower case Greek letter sigma is written s. You should read å_{j = 3}⁶f(j) as the sum from j=3 to j=6 of f of j.

Similarly, mathematical shorthand for the sum

S = 22+32+42+52+62+72

is S = å_{j = 2}⁷ j². The latter may be read as the sum from j=2 to j=7 of j squared.

The sum of the first 150 whole numbers 1,2,3,4, ..., 150 may now be written as S = å_{j = 1}¹⁵⁰ j or as 1+2+3+4+5+6+7+¼+145+146+147+148+149+150. You can clearly see which choice gives the least amount of writing. More examples follow.

1. The sum of the first 5 squares is

   12+ 22+32+42+52 = 5 å j = 1  j2

 

2. The sum of the first 4 cubes is

   13+ 23+33+43 = 5 å j = 1 j3

3. The sum of the first 3 odd numbers is

   1+3+5 = [2(0)+1]+[2(1)+1]+[2(2)+1] = 2 å j = 0  2j+1

4. The sum of the first 3 odd numbers also equals

   1+3+5 = [2(1)-1]+[2(2)-1]+[2(3)-1] = 3 å j = 1  2j-1

5. The sum of the first 4 even numbers

   2+4+6+8 = 2(1)+2(2)+2(3)+2(4) = 4 å j = 1  2j

6. The horrendous sum

   5 å k = 3  k+1 k+3  =   3+1 3+3 + 4+1 4+3 + 5+1 5+3  =  4 6 + 5 7 + 6 9

   The expression

   4 6 + 5 7 + 6 9  =  2 3 + 5 7 + 2 3  =  4 3 + 5 7  =  28+15 21  =  43 21  = 2+ 1 21

   In this sum, giving the summation notation first seemed simpler than trying to describe the sum and its terms with words only.

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