# Maths Essential formulas for competitive Exams

### ALGEBRA:

1. Sum of first n natural numbers = n(n+1)/2

2. Sum of the squares of first n natural numbers = n(n+1)(2n+1)/6

3. Sum of the cubes of first n natural numbers = [n(n+1)/2]^2

4. Sum of first n natural odd numbers = n^2

5. Average = (Sum of items)/Number of items

### Arithmetic Progression (A.P.):

An A.P. is of the form a, a+d, a+2d, a+3d…

Where a is called the ‘first term’ and d is called the ‘common difference’

1. nth term of an A.P. tn = a + (n-1)d

2. Sum of the first n terms of an A.P. Sn = n/2[2a+(n-1)d] or Sn = n/2(first

term + last term)

### Geometrical Progression (G.P.):

A G.P. is of the form a, ar, ar2, ar3…

Where a is called the ‘first term’ and r is called the ‘common ratio’.

1. nth term of a G.P. tn = arn-1

2. Sum of the first n terms in a G.P. Sn = a|1-rn|/|1-r|

### Tests of Divisibility:

1. A number is divisible by 2 if it is an even number.

2. A number is divisible by 3 if the sum of the digits is divisible by 3.

3. A number is divisible by 4 if the number formed by the last two digits is

divisible by 4.

4. A number is divisible by 5 if the unit digit is either 5 or 0.

5. A number is divisible by 6 if the number is divisible by both 2 and 3.

6. A number is divisible by 8 if the number formed by the last three digits is

divisible by 8.

7. A number is divisible by 9 if the sum of the digits is divisible by 9.

8. A number is divisible by 10 if the unit digit is 0.

9. A number is divisible by 11 if the difference of the sum of its digits at odd

places and the sum of its digits at even places, is divisible by 11.