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Introduction |
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A set of numbers arranged in a definite order according to some definite rule is called a sequence. |
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Sequences and Series |
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A set of numbers arranged in a definite order according to some definite rule is called a sequence. A sequence is a function whose domain is the set N of natural numbers. |
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Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series. |
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Arithmetic Progression |
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Quantities are said to be in Arithmetic progression when they increase or decrease by a common difference. |
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Some Properties of A.P. |
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If a,b,c,d are in A.P., then |
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(ii) ka, kc, kb, kd …are also in A.P. |
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Arithmetic Mean (A.M.) |
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If a, x, b are in A.P, then x is called the arithmetic mean (A.M.) between the extremes a and b. |
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Geometric Progressions (G.P.) |
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The series of terms a, ar, ar2, ar3,.... in which each term bears a constant ratio to the preceeding term is a geometric progression. The constant ratio is called the common ratio. |
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Harmonic Progression (H.P.) |
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A sequence of numbers is said to form a harmonic progression if their reciprocals form an arithmetic progression. |
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Harmonic Mean (H.M.) |
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If three quantities are in harmonic progression, then the middle quantity is called the harmonic mean between the other two. |
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To insert n Harmonic Means between two given quantities |
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Let a and b be two given quantities. It is required to insert n harmonic means h1, h2, h3,....hn between the quantities a and b. |
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Arithmetic Geometric Series |
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A series of the form a + (a + d)r + (a + 2d)r2 + ... is called an Arithmetic-Geometric series. In the series if we put d = 0 we get GP and if we put
r = 1, we get an AP. |
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Sigma Notation |
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The Greek letter S (read as sigma) denotes the sum. When written before the nth term of series, implies, the sum of all terms obtained by giving to n the different values 1,2,3…n. |
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General Series |
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1. To find the sum of first n natural numbers.
2. To find the sum to squares of first n natural numbers.
3. To find the sum to the cubes of first n natural numbers.
4. Method of finding sum of a series whose nth term is known. |
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Summary |
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Let X be a set of numbers and f : Nn
--> X be a function, then the ordered set {f(1), f(2),...., f(n)} is called a finite sequence in X.
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