# Geometric Progression (G. 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
• Nth Term of Geometric Progression (G.P.)
• General Term of a Geometric Progression (G.P.)
• Sum of First N Terms of a Geometric Progression (G.P.)
• Sum of infinite terms of a G.P.
• Geometric Mean (G.M.)

## Notes

Let us consider the following sequences:  2,4,8,16,...,
we have a_1 = 2 , a_2/a_1 = 2 , a_3/a_2 = 2, a_4/a_3 = 2 and so on.
In above sequence the constant ratio is 2.  Such sequences are called geometric sequence or geometric progression abbreviated as G.P.

1) General term of a G .P:
Let us consider a G.P. with first non-zero term ‘a’ and common ratio ‘r’. The second  term is obtained by multiplying a by r, thus a_2 = ar. Similarly, third term is obtained by multiplying a_2 by r. Thus, a_3 = a_2r = ar^2, and so on.
The nth term of a G.P. is given by a_n =ar^(n-1).
The series a + ar + ar^2 + ... + ar^(n–1)or a + ar + ar^2 + ...+ ar^(n–1) +... are called finite or infinite geometric series, respectively.

2)  Sum to n terms of a G .P.:
The first term of a G.P. be a and the common ratio be r.
s_n =(a(1-r^n))/1-r        or  s_n = (a(r^n -1))/r-1

3)  Geometric Mean (G .M.):
The geometric mean of two positive numbers a
and b is the number sqrt (ab) .
G_1, G_2,…, G_n be  n numbers between positive numbers a and b such that a,G_1,G_2,G_3,…,G_n, b is a G.P.

G_n =ar^n =a (b/a)^(n/(n + 1))

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