1) The document discusses different types of numerical sequences: arithmetic progressions where each term increases by a constant difference, geometric progressions where each term is multiplied by a constant ratio, and harmonic progressions where the reciprocals of the terms form an arithmetic sequence. 2) It provides the formulas for calculating the nth term of each type of progression and gives examples of sequences that fit each pattern. 3) Additionally, it briefly covers symmetric functions, which remain unchanged when two variables are swapped, and provides formulas for calculating symmetric polynomials involving the roots of polynomial equations.

1. CLASSICAL ALGEBRA
D.ARIVUSELVI

2. SYNOPSIS
• Arithmetic progression
• Geometric progression
• Harmonic progression

3. ARITHMETIC PROGRESSION
T
• A sequence of the form,
a,a+d,a+2d,a+3d,......a+(n-1)d,a+nd,....
is called an arithmetic progression (or) sequence.
Here, a=First term
d=Common difference
• The n th term of an arithmetic progression is given by,
dnaTn )1( 

4. EXAMPLES
•
,.....332,322,32,2 
Common difference=
12 , 9 , 6 , 3 ,........................................
Common difference = -3
3

5. GEOMETRIC PROGRESSION
• A sequence of the form,
with a≠ 0, r ≠ 0 is called a geometic progression (or) sequence .
Here r = Common ratio ,
a=First term
The n th term of a geometric progression is given by,
,....,,....,,,, 132 nn
ararararara 
1
 n
n arT

6. EXAMPLES
,....16,24,4,22,2,2
• 1 , 2 , 4 , 8 , 16,.......
Common ratio=2
Common ratio = 2

7. HARMONIC PROGRESSION
A sequence h1 , h2 , h3,........is said to be a harmonic sequence (or
)harmonic progression if is an arithmetic sequence
The general harmonic mean will be of the form
......
4
1
,
3
1
,
2
1
,
1
1
hhhh
,......
3
1
,
2
1
,
1
,
1
dadadaa 

8. EXAMPLES
is a harmonic sequence.
It is the reciprocal of Arithmetic progression.
.,.........
3
1
,
2
1
,1

9. SYMMENTRIC FUNCTION
If a function involving all the roots
of an equation is unaltered in
value any two of the roots are
interchanged it is called a
symmetric function of the roots

10. FORMULAE
• When the higher degree is 3,
   2)( 22
 3))((2


11. • When the higher degree is 4,
   2)( 22
   62)( 2222
    4))((2

12.     223
)()(

13. END