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: Arithmetic Sequences and Sums

3  Geometric Sequences and Sums

A sequence of numbers c₁,c₂,c₃,c₄,¼ is called an geometric sequence if there is a constant a such that multiplying each term in the sequence by a yields the next term in the sequence. That is, c_j+1 = c_j·a for some constant multiplier a which does not depend on j. An example of an arithmetic sequence is given by 6, 12, 24, 48, 96, 186 and so on. The j-th term in this sequence is given by c_j = 3·2^j. The constant multiplier here is a = 2. Three more examples follow:

1,2,4,8,16,32,64,128, ¼ Multiplier a = 2 4,2,1, 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 64 , 1 128 ,¼ Multiplier a = 1 2 . 4,4·31,4·32,4·33,4·34,4·35, ¼ Multiplier a = 3

Except for the first term, each term in a geometric sequence is a constant multiple a of the previous term. Geometric sequences occur in many money computations. Observe if c_k = f(k) = a^k·R = (1+i)^k P for k = 0,1,2,3, ¼ then the sequence c_k forms an geometric sequence with initial element c₀ = a⁰R = R. (Instead of talking about sequences a₁,a₂,a₃,¼, we can talk about sequences a_m,a_m+1,a_m+3,¼ where m is any integer. Sequences in general do not have to start with m = 1. )

3.1  Geometric Sums

The sum of terms from a geometric sequence is called a geometric sum. Just as there is an addition formula for adding arithmetic sequences, there is also an addition formula for the sum of a geometric sequence.

Example:   Suppose we want to compute

S = 7+7·51+7·52+7·53+7·54

The five numbers in this sum form a geometric sequence with multiplier a = 5. We will subtract S from 5S = aS. The three dots _.^._. is the shorthand symbol for the word therefore.

aS =      +7·51+7·52+7·53+ 7·54+7·55 -S = -7-7·51-7·52-7·53-7·54       ...  aS-S = -7+7·55

This yields (a-1)S = aS-S = 7·(5⁵-1) = a⁵-1. Thus

S = 7· a5-1 a-1 = 7· 55-1 5-1

The multiplier a is kept in this calculations to indicate the pattern that is followed. Therefore S = 7·[(3125-1)/(5-1)] = 7·[3124/4] = 7·781 = 5467.

Another Example:   Compute

S = 3+3· æ ç è 2 3 ö ÷ ø 1   +3· æ ç è 2 3 ö ÷ ø 2   +3· æ ç è 2 3 ö ÷ ø 3

The four numbers in this sum form a geometric sequence with multiplier a = ([2/3]). Again we subtract S from aS = 3S. The calculation follows.

aS =        3· æ ç è 2 3 ö ÷ ø 1   +3· æ ç è 2 3 ö ÷ ø 2   +3· æ ç è 2 3 ö ÷ ø 3   +3· æ ç è 2 3 ö ÷ ø 4   -S = -3-3· æ ç è 2 3 ö ÷ ø 1   -3· æ ç è 2 3 ö ÷ ø 2   -3· æ ç è 2 3 ö ÷ ø 3          ...     aS-S = -3+3· æ ç è 2 3 ö ÷ ø 4

This implies (a-1)S = 3·(([2/3])⁴-1) = 3·(a⁴-1). Therefore

S = 3· a4-1 a-1 = 3· æ ç è 2 3 ö ÷ ø 4   -1 2 3 -1 = 3· 1- æ ç è 2 3 ö ÷ ø 4   1- 2 3

Simplification yields S = [65/9] = 7+[2/9].

The general pattern is as follows. Suppose we want to compute

S = R+R·a1+R·a2+¼+R·am-1+R·am

The five numbers in this sum form a geometric sequence with multiplier a. We will subtract S from aS.

aS =       +R·a1+R·a2+¼+R·am+R·am+1 -S = -R-R·a1-R·a2-¼-R·am       ...     aS-S = -R+R·am+1

This implies (a-1)S = R·(a^m+1-1). Thus a ¹ 1 implies

S = R· am+1-1 a-1

When m is large, this formula for S is quicker to calculate than the initial formula

S = R+R·a1+R·a2+¼+R·am-1+R·am

Exercise.   Write out or think through the above argument without the three dots ¼ notation for the three cases m = 4, m = 3 and m = 2.

A Special Case. The case where R = P·aⁿ is often of interest in money calculations. In this case, the geometric sum

S = Pan+P·an+1+P·an+2+¼+P·an+m-1+P·an+m

yields the same result as

S = P·an· am+1-1 a-1

Remark. Geometric sums are useful in calculations involving debts, loans, pension plans and mortgages when interest is compounded. Geometric sums can be used to find the limiting value of repeating decimal expansions. Geometric sums also appear in the discussion of convergence of polynomial approximations or representation of functions and formulas. In particular, geometric sums also appear in the polynomial approximation of logarithms. Geometric sums play a key role in the arithmetic-based perspective of higher mathematics.

Next Chapter: -23 Summation and Product Notation

www.whyslopes.com
Volume 2, Three Skills for Algebra -

Preview, starter & further lessons for logic and algebra to (i) improve work & study skills;  (ii) to  to ease or avoid algebra (math) fears & difficulties; and (iii) to fill gaps in the exposition of mathematics.

Foreword, Chapters and Appendices follow.

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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