# Concept of Sequences

#### Topics

• ##### Sequences and Series
• Concept of Sequences
• Geometric Progression (G.P.)
• General Term Or the nth Term of a G.P.
• Sum of the First n Terms of a G.P.
• Sum of Infinite Terms of a G. P.
• Recurring Decimals
• Harmonic Progression (H. P.)
• Types of Means
• Special Series (Sigma Notation)
• ##### Locus and Straight Line
• Locus
• Equation of Locus
• Line
• Equations of Lines in Different Forms
• General Form Of Equation Of Line
• ##### Determinants
• Determinants
• Properties of Determinants
• Application of Determinants
• Cramer’s Rule
• Consistency of Three Linear Equations in Two Variables
• Area of a Triangle Using Determinants
• Collinearity of Three Points
• ##### Limits
• Definition of Limit of a Function
• Algebra of Limits
• Evaluation of Limits
• Direct Method
• Factorization Method
• Rationalization Method
• Limits of Exponential and Logarithmic Functions
• ##### Continuity
• Continuous and Discontinuous Functions
• Continuity of a Function at a Point
• Definition of Continuity
• Continuity from the Right and from the Left
• Properties of Continuous Functions
• Continuity in the Domain of the Function
• Examples of Continuous Functions Whereever They Are Defined
• ##### Differentiation
• The Meaning of Rate of Change
• Definition of Derivative and Differentiability
• Derivative by the Method of First Principle
• Rules of Differentiation (Without Proof)
• Applications of Derivatives
• ##### Partition Values
• Concept of Median
• Partition Values
• Quartiles
• Deciles
• Percentiles
• Relations Among Quartiles, Deciles and Percentiles
• Graphical Location of Partition Values
• ##### Measures of Dispersion
• Measures of Dispersion
• Range of Data
• Quartile Deviation (Semi - Inter Quartile Range)
• Variance and Standard Deviation
• Standard Deviation for Combined Data
• Coefficient of Variation
• ##### Skewness
• Skewness
• Asymmetric Distribution (Positive Skewness)
• Asymmetric (Negative Skewness)
• Measures of Skewness
• Karl Pearson’S Coefficient of Skewness (Pearsonian Coefficient of Skewness)
• Features of Pearsonian Coefficient
• Bowley’s Coefficient of Skewness
• ##### Bivariate Frequency Distribution and Chi Square Statistic
• Bivariate Frequency Distribution
• Classification and Tabulation of Bivariate Data
• Marginal Frequency Distributions
• Conditional Frequency Distributions
• Categorical Variables
• Contingency Table
• Chi-Square Statistic ( χ2 )
• ##### Correlation
• Correlation
• Concept of Covariance
• Properties of Covariance
• Concept of Correlation Coefficient
• Scatter Diagram
• Interpretation of Value of Correlation Coefficient
• ##### Permutations and Combinations
• Introduction of Permutations and Combinations
• Fundamental Principles of Counting
• Concept of Multiplication Principle
• Concept of Factorial Function
• Permutations
• Permutations When All Objects Are Distinct
• Permutations When Repetitions Are Allowed
• Permutations When All Objects Are Not Distinct
• Circular Permutations
• Properties of Permutations
• Combination
• Properties of Combinations
• ##### Commercial Mathematics
• Percentage
• Profit and Loss
• Simple and Compound Interest (Entrance Exam)
• Depreciation
• Partnership
• Goods and Service Tax (GST)
• Shares and Dividends
• Finite sequence
• Infinite sequence
• Progression

## Notes

Let us consider the following examples: Assume that there is a generation gap of 30 years, we are asked to find the number of ancestors, i.e., parents, grandparents, great grandparents, etc. that a person might have over 300 years.
Here, the total number of generations = 300 /30 =10
The number of person’s ancestors for the first, second, third, …, tenth generations are 2, 4, 8, 16, 32, …, 1024. These numbers form what we call a sequence.
The n^(th) term is the number at the nth position of the sequence tand is denoted by a^n.The n^(th) term is also called the general term of the sequence.
A sequence containing finite number of terms is called a finite sequence. For example, sequence of ancestors is a finite sequence since it contains 10 terms (a fixed number).
A sequence is called infinite, if it is not a finite sequence.  For example, the sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends.
a sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it. Sometimes, we use the functional notation a(n) for a_n.

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