1. C OUNTING A DD S UMMATION An Introduction

2. C OUNTING Here counting doesn’t mean counting the things physically. Here we will learn, how we can count without counting. Two basic things regarding counting are: 1. Sequences 2. Series

3. S EQUENCE Let N be the set of natural numbers and N n be the set of first n natural numbers, i.e., N n ={1,2,3,4,……….n} and X be a non-empty set, then a map f:N n  X is called a finite sequence and a map f:N  X is called an infinite sequence. A sequence following some definite rule (or rules) is called a “progression”.

4. Illustrations: 3,5,7,9,……………….,21 8,5,2,-1,-4,………..,16 2,6,18,54,………,1458 1,4,9,16,……………… 1,3,5,7,9,…………….. 8,5,2,-1,-4,……….. } Finite sequence } Infinite sequence

5. If the terms of sequence are connected by plus (or minus) signs, a series is formed. Thus, if T n denotes the general term of a sequence, then T 1 + T 2 + T 3 + T 4 +………+ T n is a series of n terms. S ERIES

6. Illustrations: 3+5+7+9+…………+21 8+5+2+1+4+………+16 2+6+18+54+………+14 58 1+4+9+16+…………. 1+3+5+7+9+………... } Finite Series } Infinite Series

7. Arithmetic Progression(A.P.) : A sequence (finite or infinite) is called an arithmetic progression ( abbreviated A.P.). Iff the difference of any term from its preceding term is constant. This constant is usually denoted by d and is called common difference. The first term of A.P. is usually denoted by a. for example: The sequence 3,5,7,9,…..21 is a finite A.P. with d=2 and the sequence 8,5,2,-1,…….. Is an infinite A.P. with d= - 3 A RITHMETIC P ROGRESSION

8. Let a be the first term and d the common difference of an A.P. let T 1, T 2, T 3,…………. T n denote 1 st, 2 nd,3 rd,……. n th terms respectively, then we have. T 2 – T 1 = d T 3 – T 2 = d T 4 – T 3 = d ……………. ……………. T n – T n-1 = d General Term= T n = a+(n-1)d G ENERAL TERMS OF A.P.

9. If l is the last term of A.P., then the number of terms in the A.P. is n= l-a+d /d And the common difference d= l-a/n- 1 IIf a 1, a 2, a 3,………..,a n are non-zero numbers such that 1/ a 1,1/ a 2,1/ a 3,…………, 1/ a n are in A.P., then a 1, a 2, a 3,………,a n are said to be in “ Harmonic Progression” (abbreviated H.P.) l-a d= --------- n- 1 l-a d= --------- n- 1 l-a+d n= -------- d l-a+d n= -------- d

10. S UM OF N TERMS OF AN A.P. let a be the first term, d the common difference and l the last term of the given A.P. then S n, sum of n terms of this A.P. can be calculated by three different mechanism depending upon input values : 1. If a, n and d is given 2. If a, n and l is given 3. If n, l and d is given S n = n/2{2 l -(n-1)d}

11. A RITHMETIC M EANS A.M. When three numbers are in A.P., the middle one is said to be the Arithmetic Mean between the other two. if a,b and c are in A.P., then A.M.= b To find the A.M. between two given numbers A= a+b/2 where a and b are two given numbers. if we have n terms between a and b then A= n{(a+b)/2}

12. G EOMETRIC P ROGRESSION G.P. A sequence (finite or infinite) of non-zero terms is called a “geometric progression” iff the ratio of any terms to its preceding term is constant. this (non-zero) constant is usually denoted by r and is called “Common ration” We assume that none of the terms of the sequence is zero. Eg. a,ar,ar 2, ar 3, ar 4,…………., ar n-1 is in G.P.

13. Let a be the first term and r be the common ratio of a G.P. let T 1, T 2, T 3,…………. T n denote 1 st, 2 nd,3 rd,……. n th terms respectively, then we have. T 2 = T 1 r T 3 = T 2 r T 4 =T 3 r ……………. ……………. T n =T n-1 r General Term= T n = ar n-1 G ENERAL TERMS OF G.P.

14. Sum of n terms of a G.P. depends upon the value of r. there are three possibilities as: 1. If r <1 2. If r ≤ -1 or r >1 3. If r=1 S UM OF N TERMS OF A G.P.

15. S UMMATION