MATHS FORMULAS -Number system

 MATHS FORMULAS

NUMBERS :

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

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# 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)

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# 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|

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# Permutations and Combinations :

nPr = n!/(n-r)!
nPn = n!
nP1 = n
nCr = n!/(r! (n-r)!)
nC1 = n
nC0 = 1 = nCn
nCr = nCn-r
nCr = nPr/r!
Number of diagonals in a geometric figure of n
sides = nC2-n

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# Tests of Divisibility :

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

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

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

* A number is divisible by 5 if the units digit is
either 5 or 0.

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

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

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

* A number is divisible by 10 if the units digit is 0.

* 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.

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# H.C.F and L.C.M :

H.C.F stands for Highest Common Factor. The
other names for H.C.F are Greatest Common
Divisor (G.C.D) and Greatest Common Measure
(G.C.M).
* The H.C.F. of two or more numbers is the
greatest number that divides each one of them
exactly.
* The least number which is exactly divisible by
each one of the given numbers is called their
L.C.M.
* Two numbers are said to be co-prime if their
H.C.F. is 1.
* H.C.F. of fractions = H.C.F. of numerators/
L.C.M of denominators
* L.C.M. of fractions = G.C.D. of numerators/
H.C.F of denominators
Product of two numbers = Product of their
H.C.F. and L.C.M.