Algebra Formulas

MathsIsFun · 2006-03-29 10:07:36

Algebra Formulas

Jai Ganesh · 2006-03-30 03:33:51

Jai Ganesh · 2006-04-02 01:13:38

A quadratic equation consists of a single variable of degree 2 and is of the form

The roots of the equation are given by

Two roots or solutions are obtained, but sometimes they may be equal. If the discriminant b²-4ac>0, the roots are real and distinct. If b²-4ac=0, the roots are real and equal. If b²-4ac<0, the roots are distinct and imaginary.

The sum of the roots = -b/a

Product of the roots = c/a

Given the roots of the quadratic equation, the quadratic can be formed using the formula
x²-(sum of the roots)x + (product of the roots)=0.

Jai Ganesh · 2006-04-04 03:52:24

I. Arithmetic Progressions.
An Arithmetic Progression (AP) is a series in which the succesive terms have a common difference. The terms of an AP either increase or decrease progressively. For example,
1, 3, 5,7, 9, 11,....
10, 9, 8, 7,6, 5, .....
14.5, 21, 27.5, 34, 40.5 .....
11/3, 13/3, 15/3, 17/3, 19/3......
-5, -8,-11, -14, -17, -20 ......
Let the first term of the AP be a and the common difference, that is
the difference between any two succesive terms be d.

The nth term, tn is given by

The sum of n terms of an AP, Sn is given by the formula

or

where l is the last term (nth term in this case) of the AP.

Jai Ganesh · 2006-04-04 03:58:16

II. Geometric Progression

a, b, c, d, ... are said to be in Geometric Progression (GP) if
b/a = c/b = d/c etc.

A Geometric Progression is of the form

etc.
where a is the first term and r is the common ratio.

The nth term of a Geometric Progression is given by

The sum of the first n terms of a Geometric Progression is given by
(i) When r<1

(ii) When r>1

Sum of the infinite series of a Geometric Progression when |r|<1

Geometric Mean (GM) of two numbers a and b is given by

Jai Ganesh · 2006-04-06 02:09:05

Harmonic Progression:-

A Harmonic Progression (HP) is is a series of terms where the reciprocals of the terms are in Arithmetic Progression (AP).

The general form of an HP is
1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), .....

The nth term of a Harmonic Progression is given by
tn=1/(nth term of the corresponding AP)

In the following Harmonic Progression

The Harmonic Mean (HM) of two numbers a and b is

The Harmonic Mean of n non-zero numbers

is

Relation between Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM)

that is, AM, GM, HM are in Geometric Progression.

For two positive numbers,
AM ≥ GM ≥ HM equality holding for equal numbers.

For n non-zero positive numbers, AM ≥ GM ≥ HM

Jai Ganesh · 2006-04-06 02:28:14

SUMMATION

Jai Ganesh · 2006-04-18 03:45:51

Laws of Exponents

In all the above cases,

where a is a non-zero real number.

and n is a non-negative number.

If a is a postive real number and m,n are integers with n positive,

If and b are positive real numbers and n a natural number, then

If

, then a=b.

If

then m=n.

Jai Ganesh · 2006-05-03 02:22:33

Binomial theorem

If n is a positive integer,

where

Summation of Binomial coefficients

If n is a rational index and -1<x<1, then

Some expansions:-

Jai Ganesh · 2006-05-10 04:05:40

Expansions of Logarithmic expressions

Jai Ganesh · 2006-05-11 03:48:23

AP, GP, HP :- Some important results

If A is the Arithmetic Mean (AM) of two numbers a and b, and G is their Geometric Mean (GM), then the two numbers are given by

For example, let the two numbers be 4 and 16. The AM of the two numbers is 10 and their GM is 8.
Therefore, A=10, G=8

gives two values, viz. 16 and 4.

If

are in Geometric Progression, then
are in Arithmetic Progression.

Similarly, if

are in Arithmetic Progression, then

are in Geometric Progression.

Jai Ganesh · 2006-05-14 00:52:28

Arithmetico-Geometric Series

A series having terms
a, (a+d)r, (a+2d)r², .... etc. is an Arithmetico-Geometric series where a is the first term, d is the commom difference of the Arithmetic part of the series and r is the common ratio of the Geometric part of the series.
An example of Arithmetico-Geometric series is
10, 9/2, 2, 7/8, 3/8, 5/32.... wherea=10, d=-1, and r=1/2.

The nth term

In the above example, the third term is[a+2d]r², i.e.2.

The sum of the series to n terms is

In the series given above, the sum of the first four terms would be
20+[(-1/2)(7/8)][1/4]-7(1/16)/(1/2)=20-7/4-7/8=20-21/8=139/8.

It can be seen, 10+9/2+2+7/8=(80+36+16+7)/8=139/8.

The sum to infinity,

Devantè · 2006-10-10 00:26:06

$d20d39c458e7add6ca7b0557dde16e1.gif$

MissK · 2007-02-28 16:13:49

This is all very handy to a 4th grade teacher who took calculus only 25 short years ago! My current problem is to find out a formula for what another web site calls a "triangular number" pattern. 1, 3, 6, 10, 15 . . .

More to the point, how do I find all possible UNIQUE triple-dip combinations of ice cream cones. I know the formula for double-dips, but darned if I can't figure this one out.

Any help?? I am humbled by all of your brilliances.

Thank you -- K

Jai Ganesh · 2007-02-28 16:25:48

Hi MissK,
Welcome to the forum and thanks for posting what you think of the forum!
The series 1, 3, 6, 10, 15 has the first term (lets call it 'a' for convenience) as 1 and thereafter, the difference between the nth term and (n-1)th term is n. That is, the difference between the 3rd and 2nd term is 3, the difference between the 4th and the third terms is 4 and so on. Thus, the 2nd term is 2+a, the third term is 3+(2+a), the fourth term is 4+[3+(2+a)]. It can be seen that the nth term is

.

They form a series of numbers which are the sum of the first n natural numbers.

MathsIsFun · 2007-03-01 00:00:16

Triangular Numbers: Definition of Triangular Number

Combinations: Combinations and Permutations

Hope they help!

(But we may need to delete this conversation as it is in the formulas )

MissK · 2007-03-01 15:25:08

MathsIsFun wrote:

Triangular Numbers: Definition of Triangular Number
Combinations: Combinations and Permutations
Hope they help!
(But we may need to delete this conversation as it is in the formulas )

Oh -- Oops! Sorry. That means I can only write brilliant formulas. OK
Thanks for the tolerance. Delete.

lightning · 2007-03-07 20:20:03

'im learning a lot here

R3hall · 2007-05-09 15:01:55

An example of figurometry formulas:
N2 = square number
√ = square root
Tn = triangle number
(-1+√(8n+1))/2= triangle root

simron · 2008-04-18 12:53:12

(a+b)^c = sum starting at k=1 to infinity((v choose k) x^k a^(v-k))
I couldn't get that to work in LaTeX.

Jai Ganesh · 2009-01-02 03:06:20

Componendo-Dividendo

If

, then,

which is called the componendo.

If

, then,

which is called dividendo.

If

, then,

which is called the componendo & dividendo.

Jai Ganesh · 2009-01-10 01:30:24

1.

2. If a+b+c = 0,

3. (i)

is divisible by (x-a) for all values of n.

3. (ii)

is divisible by (x+a) for all even values of n.

3.(iii)

is divisible by (x+a) for all odd values of n.

Daniel123 · 2009-01-10 02:41:49

A useful algebraic identity:

JaneFairfax · 2009-01-10 02:51:39

This is also useful.

[Dickinson]a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2\\\\
=\ (a+b)(b+c)(c+a) - 2abc\\\\
=\ (a+b+c)(ab+bc+ca) - 3abc[/Dickinson]

And this.

[Dickinson]a^3 + b^3 + c^3\ =\ (a + b + c)^3 - 3(a + b + c)(ab + bc + ca) + 3abc[/Dickinson] (slightly different from Ganeshs formula)

Denominator · 2009-11-24 06:18:43

I learned limits today!!!!

I love Algebra and Calculus!!

I hate estimating and probability = [

Math Is Fun Forum

#1 2006-03-29 10:07:36

Algebra Formulas

#2 2006-03-30 03:33:51

Re: Algebra Formulas

#3 2006-04-02 01:13:38

Re: Algebra Formulas

#4 2006-04-04 03:52:24

Re: Algebra Formulas

#5 2006-04-04 03:58:16

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#6 2006-04-06 02:09:05

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#7 2006-04-06 02:28:14

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#8 2006-04-18 03:45:51

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#9 2006-05-03 02:22:33

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#11 2006-05-11 03:48:23

Re: Algebra Formulas

#12 2006-05-14 00:52:28

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#13 2006-10-10 00:26:06

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#14 2007-02-28 16:13:49

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#15 2007-02-28 16:25:48

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#16 2007-03-01 00:00:16

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#17 2007-03-01 15:25:08

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#18 2007-03-07 20:20:03

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#19 2007-05-09 15:01:55

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#20 2008-04-18 12:53:12

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#21 2009-01-02 03:06:20

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#22 2009-01-10 01:30:24

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