Algebra Formulas
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.
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.
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
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
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
SUMMATION
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.
