What is the difference between sequence and series?

When some terms are arranged in a definite order in accordance with some rule, it is called Sequence.
For example: 1,3,5,7,9 is a sequence of 5 terms containing consecutive odd numbers.
Now, the sum of a particular sequence is known as series.
Here, 1+3+5+7+9 is the sum of the particular sequence.

How are the terms and sum of a sequence expressed?

Generally, the "n"th term of a sequence is represented as an or Tn ,where n is a positive integer.
Similarly, the sum of a sequence upto n terms is represented as Sn
Note. Tn= Sn - Sn-1 , that is, the "n"th term of a sequence is the difference between the sum of n terms and (n-1) terms of that sequence.
Example: If Sn= n2+3n+2, find T3
Here, n=3, thus all we need to find out sum of n=3 terms and sum of n-1=2 terms.
S3= (3)+3(3)+2=20 and S2=(2)+3(2)+2=12
Hence, T3=S3-S2=20-12=8

Types of Sequences

1. Arithmetic progression (AP)

It is a sequence in which the difference between a term and its previous term is constant throughout the sequence. The first term in this sequence is represented as 'a' while the difference is represented as 'd'(also called common difference)
The 'n'th term is thus represented as, Tn= a+(n-1)d ,
While the sum of n terms is shown as Sn=(n/2)*[2a+(n-1)d] = (n/2)[a+l] where l
is the 'n'th term of the AP.
Note.
• d= Tn - Tn-1
• 3 numbers are taken in AP as a-d, a , a+d
• 4 numbers in AP are taken as a-3d ,a-d ,a+d ,a+3d
• If n numbers x1,x2, ... xn are given(not AP), Arithmetic mean(AM) is shown as (x1+x2 ...+xn)/n
• If a,b,c are in AP, then AM=(a+c)/2=b

2. Geometric Progression (GP)

It is a sequence in which the ratio between the term and its previous term is a constant throughout the sequence. The first term is represented as 'a' while
the ratio is called 'common ratio' and represented as 'r'
The 'n'th term is represented as Tn=a*rn-1
The sum of n terms is shown as S= a(rn-1)/(r-1)
If n tends to infinite, or the number of terms is infinite, then S=a/(1-r)
Note.
• r=Tn/Tn-1
• 3 numbers in GP are taken as a/r ,a ,ar
• 4 numbers in GP are taken as a/r3 , a/r , ar ,ar3
• If a,b,c are in GP, then Geometric mean(GM)= (a*c)0.5
• If n numbers x1,x2, ... xn are given(not GP), GM is shown as (x1 *x2 * ... xn)1/n

3. Harmonic Progression (HP)

It is a sequence in which the reciprocals of the terms form an arithmatic progression.
Thus, Tn= 1/[a+(n-1)d] where a and d are the first term and common difference of the Specific AP.
Note.
• If a,b,c are in HP, Harmonic mean(HM)= 2ab/a+b
• If n numbers x1,x2, ... xn are given(not HP), HM is shown as n/[(1/x1)+(1/x2) ... +(1/xn)]
Some important formulae
1. 1+2+3+........n= n(n+1)/2
2. 12+22+32+........n2= n(n+1)(2n+1)/6
3. 13+23+33........n3= {n2(n+1)2 }/4
NOTE.
AM=GM=HM always stands true and AM=GM=HM holds true only when all the
terms are equal in magnitude.