Sequences and Series

A sequence is an ordered array of numbers which follow a particular rule. A sequence can be defined as a function whose domain is the set of natural numbers or some subset of it of the type {1, 2, 3...k}. a(n) or an represents a sequence. Various numbers occurring in the sequence are called the terms of the sequence. A sequence having the finite number of terms is called the finite sequence. A sequence having the infinite number of terms is called the infinite sequence. For the given sequence a1, a2, a3, …, an, …, the series associated with this sequence is given by the expression a1 + a2 + a3 + … + an + … A sequence is called an arithmetic progression (A.P.) if each term except the first term is obtained by adding a fixed number to the immediate preceding term. The fixed number in A.P. is called the common difference of the A.P. The general term or nth term of above AP is given by: an = a + (n – 1)d Sum of first n terms of above A.P. is given by: Sn = (n/2) × [ 2a + (n–1)d] = (n/2) × [a + ] where a is the first term,  is the last term and d is the common difference of A.P. The arithmetic mean C of any two numbers a and b is given by (a + b)/2, i.e., the sequence a, C, b is in A.P. A sequence is said to be geometric progression (G.P.) if the ratio of any term to its preceding term is constant. This constant ratio is called the common ratio. The general term or nth term of a GP is given by an = ar(n – 1) , where a is the first term and r is the common ratio of the G.P. The sum of first n terms of a G.P. is given by: Sn = [a(1 – rn) / (1 – r)] if r ≠ 1. The geometric mean G of any two numbers a and b is given by square root of a × b i.e., the sequence a, G, b is in G.P. The arithmetic mean is always greater than or equal to the geometric mean of any two positive real numbers.

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