Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

Login

Username

Password

Not registered yet?

#1 2006-03-30 09:07:36

MathsIsFun
Administrator

Offline

Algebra Formulas

Algebra Formulas


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman
 

#2 2006-03-31 02:33:51

ganesh
Moderator

Offline

Re: Algebra Formulas














Character is who you are when no one is looking.
 

#3 2006-04-02 23:13:38

ganesh
Moderator

Offline

Re: Algebra Formulas

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.


Character is who you are when no one is looking.
 

#4 2006-04-05 01:52:24

ganesh
Moderator

Offline

Re: Algebra Formulas

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.


Character is who you are when no one is looking.
 

#5 2006-04-05 01:58:16

ganesh
Moderator

Offline

Re: Algebra Formulas

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


Character is who you are when no one is looking.
 

#6 2006-04-07 00:09:05

ganesh
Moderator

Offline

Re: Algebra Formulas

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


Character is who you are when no one is looking.
 

#7 2006-04-07 00:28:14

ganesh
Moderator

Offline

Re: Algebra Formulas

SUMMATION






Character is who you are when no one is looking.
 

#8 2006-04-19 01:45:51

ganesh
Moderator

Offline

Re: Algebra Formulas

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.


Character is who you are when no one is looking.
 

#9 2006-05-04 00:22:33

ganesh
Moderator

Offline

Re: Algebra Formulas

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:-






















Character is who you are when no one is looking.
 

#10 2006-05-11 02:05:40

ganesh
Moderator

Offline

Re: Algebra Formulas

Expansions of Logarithmic expressions






Character is who you are when no one is looking.
 

#11 2006-05-12 01:48:23

ganesh
Moderator

Offline

Re: Algebra Formulas

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.


Character is who you are when no one is looking.
 

#12 2006-05-14 22:52:28

ganesh
Moderator

Offline

Re: Algebra Formulas

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,


Character is who you are when no one is looking.
 

#13 2006-10-10 22:26:06

Devantè
Real Member

Offline

Re: Algebra Formulas

http://www.mathsisfun.com/forum/latex/3/f/d20d39c458e7add6ca7b0557dde16e1.gif

 

#14 2007-03-01 15:13:49

MissK
Member

Offline

Re: Algebra Formulas

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


Strength and Honor
 

#15 2007-03-01 15:25:48

ganesh
Moderator

Offline

Re: Algebra Formulas

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.


Character is who you are when no one is looking.
 

#16 2007-03-01 23:00:16

MathsIsFun
Administrator

Offline

Re: Algebra Formulas

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 sad )


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman
 

#17 2007-03-02 14:25:08

MissK
Member

Offline

Re: Algebra Formulas

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 sad )

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


Strength and Honor
 

#18 2007-03-08 19:20:03

lightning
Real Member

Offline

Re: Algebra Formulas

'im learning a lot here


Zappzter - New IM app! Unsure of which room to join? "ZNU" is made to help new users. c:
 

#19 2007-05-10 13:01:55

R3hall
Member

Offline

Re: Algebra Formulas

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

 

#20 2008-04-19 10:53:12

simron
Real Member

Offline

Re: Algebra Formulas


(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.


Linux FTW
 

#21 2009-01-03 02:06:20

ganesh
Moderator

Offline

Re: Algebra Formulas

Componendo-Dividendo

If

, then,

which is called the componendo.

If
, then,

which is called dividendo.

If
, then,

which is called the componendo & dividendo.


Character is who you are when no one is looking.
 

#22 2009-01-11 00:30:24

ganesh
Moderator

Offline

Re: Algebra Formulas

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.


Character is who you are when no one is looking.
 

#23 2009-01-11 01:41:49

Daniel123
Power Member

Offline

Re: Algebra Formulas

A useful algebraic identity:

 

#24 2009-01-11 01:51:39

JaneFairfax
Legendary Member

Offline

Re: Algebra Formulas

This is also useful.



And this.

(slightly different from Ganesh’s formula)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#25 2009-11-25 05:18:43

Denominator
Full Member

Offline

Re: Algebra Formulas

I learned limits today!!!!

I love Algebra and Calculus!!

I hate estimating and probability = [

 

Board footer

Powered by FluxBB