General Series
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 fir..
Summary
of (a + b) n and is given by - If n is an odd natural number, then there are two middle terms in the expansion of (a + b)n and are given by i) The sum of all binomial coefficients in the expansion of (1+x) n i..
of (a + b) n and is given by - If n is an odd natural number, then there are two middle terms in the expansion of (a + b)n and are given by i) The sum of all binomial coefficients in the expansion of (1+x) n i..General Series
1. To find the sum of first n natural numbers. ..
1. To find the sum of first n natural numbers. ..Some Observations
The binomial coefficients in (a + b) n , which are equidistant from beginning and end are equal. The binomial coefficients in the expansion of (a + b) n can be easily evaluated by using the Pascal's triangle given below: In the Pascal's triangle, each row starts and en..
The binomial coefficients in (a + b) n , which are equidistant from beginning and end are equal. The binomial coefficients in the expansion of (a + b) n can be easily evaluated by using the Pascal's triangle given below: In the Pascal's triangle, each row starts and en..Sequences and Series Summary
) ka, kb, kc are in G.P. (b) a/k, b/k, c/k are in G.P. (x) (a) If the product of three numbers in G.P. is given, then the numbers should be taken as a/r, a, ar. (b) If the product of four numbers in G.P. is given, then the numbers should be taken as a/r 3 , a..
) ka, kb, kc are in G.P. (b) a/k, b/k, c/k are in G.P. (x) (a) If the product of three numbers in G.P. is given, then the numbers should be taken as a/r, a, ar. (b) If the product of four numbers in G.P. is given, then the numbers should be taken as a/r 3 , a..Introduction
A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'. For example, x - y, a + 3b, x 3 + 4y etc. are binomials. We know that, For n = 1,2,3,4, the expansion of (a + b) n , has been expressed in a very systematical..
A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'. For example, x - y, a + 3b, x 3 + 4y etc. are binomials. We know that, For n = 1,2,3,4, the expansion of (a + b) n , has been expressed in a very systematical..Matrices and Determinants
The following are the steps to solve a system of linear equations using Cramer's rule. Step 1: Find the value of the determinant Step 2: If D 0, then the system has unique solution, given by Where D 1 , D 2 and D 3 are the determinants obt..
The following are the steps to solve a system of linear equations using Cramer's rule. Step 1: Find the value of the determinant Step 2: If D 0, then the system has unique solution, given by Where D 1 , D 2 and D 3 are the determinants obt..Step I:
Denote the statement involving natural numbers n = 1, 2,. by P(n..
Sequences and Series
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. Indicated sum of the terms in a sequence is called a series. The r..
Sigma Notation
The Greek letter S (read as sigma) denotes the sum. When written before the n t h term of series, implies, the sum of all terms obtained by giving to n the different values 1,2,3n..
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