Numeral

Definition:
A group of digits,denoting a number is called numeral.
A digit represents a number using ten symbols 0,1,2,3,4,5,6,7,8,9.

A number is represented in the following manner.

Ten Crores	108
Crores	107
Ten Lakhs(Millions)	106
Lakhs	105
Ten Thousands	104
Thousands	103
Hundreds	102
Tens	101
Units	100

Example:
The number 986745123 is represented as showm below

9	Ten Crores	108
8	Crores	107
6	Ten Lakhs(Millions)	106
7	Lakhs	105
4	Ten Thousands	104
5	Thousands	103
1	Hundreds	102
2	Tens	101
3	Units	100

The number 986745123 is read as Ninety-eight crores sixty-seven lakhs forty-five thousand one hundred and twenty-three

Geometric Progression(GP)

Definition:
A progression of numbers in which every term bears a constant ratio with the preceeding term is called Geometric progression.
The constant ratio is called the common ratio of the GP
If the first term is a and common ratio is r then,
GP is a,ar,ar²,ar³,…..

Formula:
Tn = ar^n-1
Sn = (a(1-rⁿ))/(1-r)
Where,
Tn is the nth term in the GP,
Sn is the sum of n terms in the GP,
a is the first term,
r is the common ratio of the GP,
n is the number of term in the GP.

Example1:
Find the 10 th term in the GP 2,4,8,16,….
Solution:

	Step1:
		Use the formula, Tn = arn-1
	Step2:
		In the given problem,
		a=first term=2
		r=common ratio=2
		(to find common ratio take any two terms in the GP and find the quotient(4/2=2))
		n=10
	Step3:
		Apply values of step 2 in step 1
		T10=2 * 2 9
		=1024
		The 10 th term in the given GP is 1024

Example2:
Find the sum of 5 terms in the GP 2,4,8,16,….
Solution:

	Step1:
		Use the formula, Sn = (a(1-rn))/(1-r)
	Step2:
		In the given problem,
		a=first term=2
		r=common ratio=2
		(to find common ratio take any two terms in the GP and find the quotient(4/2=2))
		n=5
	Step3:
		Apply values of step 2 in step 1
		S5=(2 * (1-25))/(1-2)
		=62
		The sum of 5 terms in the given GP is 62

Geometric progression(GP)

is explained in this article
For details regarding Arithmetic Progression click here

Arithmetic Progression

Definition:
A succession of numbers formed and arranged in a definite order according to certain definite rule is called progression.
If each term of a progression differs from the preceding term by a constant,then the progression is called arithmetic progression or AP.
The constant difference is called the common difference of the AP.
If first term is a and common difference is d then,
AP is a,(a+d), (a+2d), (a+3d),……

Formula:
Consider the AP a,(a+d), (a+2d), (a+3d),……
1.Tn = a+(n-1)d
2.Sn = (n/2)(2a+(n-1)d) or
Sn = (n/2)(first term + last term)

Where,
Tn is the nth term of the AP,
Sn is the sum of n terms in the AP,
a is the first term,
n is the number of term in the AP,
d is the common difference of the AP.

Example 1:
Find the 10 th term in the AP 2,6,10,14,……
Solution:
 Step 1:use the formula Tn = a+(n-1)d
 Step 2:In the given problem,
    a = first term = 2
    d= common difference = 4
  (to find d take any two terms in the AP and subtract it ( 6-2=4))
    n = 10
 Step 3:Apply the values in the formula.
   T10=2+(10-1)4
     =2+36
    =38
 Step 4:The 10 th term in the AP is 38

Example 2:
Find the sum of 5 terms in the AP 2,6,10,….
Solution:
 Step 1:use the formula sn = (n/2)(2a+(n-1)d)
 Step 2:In the given problem,
    a = first term = 2
    d= common difference = 4
  (to find d take any two terms in the AP and subtract it ( 6-2=4))
    n = 5
 Step 3:Apply the values in the formula.
    s10=(5/2)(2*2+(5-1)4)
    =2.5 * 20
    =50
 Step 4:The sum of 5 terms in the given AP is 50.