YOU are better than YOU think. Show yourself  how:  

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Chapter 22
Geometric and Arithmetic Sums and Sequences

Previous Section: Chapter Entrance (Sequences):

2  Arithmetic Sequences and Sums

An example of an eight-member sequence is given by 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, and 7. The j-th term in this sequence is given by a_j = 3+j·[1/2] - verify this for j = 0, 1, 2, ..., 8. The difference between adjacent or successive members d = a_j+1-a_j is the constant [1/2]. According to the next definition, this eight member sequence is an arithmetic sequence.

Definition: A sequence of numbers a₁,a₂,a₃,a₄,¼ is an arithmetic sequence if and only if there is a constant d such that adding d to each term in the sequence yields the next term in the sequence. More briefly, a_j+1 = a_j+d for some constant d which does not depend on j.

2.1  Arithmetic Sums

First Example:   Suppose we want to compute the following sum

S = 11+14+17+20+23+26+29

of the seven numbers: a₁ = 11, a₂ = 14 = a₂+3, a₃ = 17 = a₂+3, a₄ = 20 = a₃+3, a₅ = 23 = a₄+3, a₆ = 26 = a₅+3 and a₇ = 29 = a₆+3. Except for the first number in this sequence, each number a_k is given by adding the constant 3 to its predecessor a_k-1, that is, the number immediately before it. To find the sum S, write it forward and backwards as follows.

The alignment of the forward and backward written sums in seven columns, count them, suggests 2S = 7(40). This implies

A Second Example:   Suppose we want to compute the following sum

S = 11+12+13+14+15+16+17+18+19

of nine numbers b₁ = 11, b₂ = 12 = b₁+1 , b₃ = 13 = b₂+1 , b₄ = 14 = b₃+1 , b₅ = 15 = b₄+1 , b₆ = 16 = b₅+1 , b₇ = 17 = b₆+1 , b₈ = 18 = b₇+1 and b₉ = 19 = b₈+1 . Except for the first number, each number is obtained by adding 1 to the number before it. A quick way to obtain the sum is to write the sum forward and backwards:

The above arithmetic sum examples follow the pattern:

2S = (Number of Terms)(First Term+Last Term)

From this pattern, we conclude the sum

S = 1 2 ·(Number of Terms)(First Term+Last Term)

Exercises. Explain why

1. the sum of the first 500 numbers 1+2+3 +¼+488 +499+500 equals 250·501.
2. the sum S = 2+4+5+6+8 is not given by

   1 2 ·(Number of Terms)(First Term+Last Term)

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2. Three Skills for Algebra 

Foreword, Chapters 
& Appendices 

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

Real Player Videos Perfect arithmetic skills with whole numbers & fractions after or besides chapters 1 to 14. Arithmetic Videos Summary Addition with Decimals Subtraction with Decimals Multiplication with Decimals Fraction Arithmetic Recognizing Primes Long Division for Decimals Square Root Simplification Greatest Common Divisors Least Common Multiples

Words Before Symbols: 
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

Complex number: starter lesson  

Solving Linear Equations:

A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.

(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v)  (½)x + 8 = 24½
(vI)  (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With Parameters

Problem Solving with Linear
Equations in one or many
unknowns, and in essentially 
one unknown - Symbols before
words. 

C. Solving Linear Eq'ns 
without
Stick Diagrams
D. Problems in 
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems

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