Before traveling a quarter, she must travel one-eighth before an eighth, one-sixteenth and so on. Before she can get halfway there, she must get a quarter of the way there. Before she can get there, she must get halfway there. Suppose Atalanta wishes to walk to the end of a path. Zeno’s Paradox is an observation which seems absurd, yet it starts sounding logically acceptable in relation to geometric sequences! Zeno’s Paradox reads:.Without considering any other changes to the reservoir’s volume, how much water will have evaporated over a one-year period? Suppose a reservoir contains an average of \(1.4\) billion gallons of water and loses water due to evaporation at a rate of \(2\%\) per month. Changes can occur to any water supply due to inflow and outflow, but evaporation is one of the factors of water depletion. Reservoirs can be the source of water supply for millions of people. ![]() `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.\) “nested” triangle, as described? The geometric mean of two positive numbers a +` `ar^(n–1) +.` are called finite or infinite geometric series, respectively. Similarly, third term is obtained by multiplying `a_2` by r. The second term is obtained by multiplying a by r, thus `a_2` = ar. with first non-zero term ‘a’ and common ratio ‘r’. Such sequences are called geometric sequence or geometric progression abbreviated as G.P. In above sequence the constant ratio is 2. Let us consider the following sequences: 2,4,8,16., Rules of Differentiation (Without Proof).Definition of Derivative and Differentiability. ![]() Limits of Exponential and Logarithmic Functions.Binomial Theorem for Negative Index Or Fraction.Middle term(s) in the expansion of (a + b)n.Binomial Theorem for Positive Integral Index.Methods of Induction and Binomial Theorem.Permutations When Some Objects Are Identical.Permutations When Repetitions Are Allowed.Permutations When All Objects Are Distinct.Algebraic Operations of Complex Numbers.Odds (Ratio of Two Complementary Probabilities).Change of Origin and Scale of Variance and Standard Deviation.Different Forms of Equation of a Circle.Locus of a Points in a Co-ordinate Plane.Area of Triangle and Collinearity of Three Points.Consistency of Three Equations in Two Variables.Minors and Cofactors of Elements of Determinants.Definition and Expansion of Determinants.Trigonometric Functions of Angles of a Triangle.Formulae for Conversion of Product in to Sum Or Difference.Formulae for Conversion of Sum Or Difference into Product.Trigonometric Functions of Triple Angle.Trigonometric Functions of Double Angles.Trigonometric Functions of Multiple Angles.Trigonometric Functions of Allied Angels.Trigonometric Functions of Sum and Difference of Angles.Domain and Range of Trigonometric Functions.Trigonometric Functions of Negative Angles. ![]()
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