# Arithmetic Progression (A.P.)

#### Topics

• ##### Angle and Its Measurement
• Directed Angle
• Angles of Different Measurements
• Angles in Standard Position
• Measures of Angles
• Area of a Sector of a Circle
• Length of an Arc of a Circle
• ##### Trigonometry - 1
• Introduction of Trigonometry
• Trigonometric Functions with the Help of a Circle
• Signs of Trigonometric Functions in Different Quadrants
• Range of Cosθ and Sinθ
• Trigonometric Functions of Specific Angles
• Trigonometric Functions of Negative Angles
• Fundamental Identities
• Periodicity of Trigonometric Functions
• Domain and Range of Trigonometric Functions
• Graphs of Trigonometric Functions
• Polar Co-ordinate System
• ##### Trigonometry - 2
• Trigonometric Functions of Sum and Difference of Angles
• Trigonometric Functions of Allied Angels
• Trigonometric Functions of Multiple Angles
• Trigonometric Functions of Double Angles
• Trigonometric Functions of Triple Angle
• Factorization Formulae
• Formulae for Conversion of Sum Or Difference into Product
• Formulae for Conversion of Product in to Sum Or Difference
• Trigonometric Functions of Angles of a Triangle
• ##### Straight Line
• Locus of a Points in a Co-ordinate Plane
• Straight Lines
• Equations of Line in Different Forms
• General Form of Equation of a Line
• Family of Lines
• ##### Conic Sections
• Double Cone
• Conic Sections
• Parabola
• Ellipse
• Hyperbola
• ##### Measures of Dispersion
• Meaning and Definition of Dispersion
• Measures of Dispersion
• Range of Data
• Variance
• Standard Deviation
• Change of Origin and Scale of Variance and Standard Deviation
• Standard Deviation for Combined Data
• Coefficient of Variation
• ##### Permutations and Combination
• Fundamental Principles of Counting
• Invariance Principle
• Factorial Notation
• Permutations
• Permutations When All Objects Are Distinct
• Permutations When Repetitions Are Allowed
• Permutations When Some Objects Are Identical
• Circular Permutations
• Properties of Permutations
• Combination
• Properties of Combinations
• ##### Methods of Induction and Binomial Theorem
• Principle of Mathematical Induction
• Binomial Theorem for Positive Integral Index
• General Term in Expansion of (a + b)n
• Middle term(s) in the expansion of (a + b)n
• Binomial Theorem for Negative Index Or Fraction
• Binomial Coefficients
• ##### Limits
• Concept of Limits
• Factorization Method
• Rationalization Method
• Limits of Trigonometric Functions
• Substitution Method
• Limits of Exponential and Logarithmic Functions
• Limit at Infinity
• ##### Continuity
• Continuous and Discontinuous Functions
• ##### Differentiation
• Definition of Derivative and Differentiability
• Rules of Differentiation (Without Proof)
• Derivative of Algebraic Functions
• Derivatives of Trigonometric Functions
• Derivative of Logarithmic Functions
• Derivatives of Exponential Functions
• L' Hospital'S Theorem

## Notes

A sequence a_1, a_2, a_3,…, an,… is called arithmetic sequence or arithmetic progression if a_(n + 1) = a_n + d, n ∈ N, where a_1 is called the first term and the constant term d is called the common difference of the A.P.
The n^(th) term (general term) of the A.P. is a^n = a + (n – 1) d.

The sum to n term of A.P is S_n= n/2[2a+(n-1)d]

We can also write, S_n = n/2[a+l]

We can verify the following simple properties of an A.P. :
(1) If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
(2) If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
(4) If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
(5) If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P.

Arithmetic mean:

Given two numbers a and b. We can insert a number A between them so that a, A, b is an A.P. Such a number A is called the arithmetic mean (A.M.) of the numbers a and b. Note that, in this case, we have
A – a = b – A,    i.e., A  =(a+b)/2
We may also interpret  the A.M. between two numbers a and b as their average (a+b)/2.
For example, the A.M. of two numbers 4 and 16 is 10. We have, thus constructed an A.P. 4, 10, 16 by inserting a number 10 between 4 and 16.
The Arithmetic mean is d = (b - a)/(n + 1)

If you would like to contribute notes or other learning material, please submit them using the button below.

### Shaalaa.com

Arithmetic Progression [00:04:26]
S
0%