Geometric Progression (G. P.)

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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
  • Determinants and Matrices
  • 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
  • Circle
  • 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
  • Probability
  • Complex Numbers
  • Sequences and Series
  • 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
  • Sets and Relations
  • Functions
  • 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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