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 = n2
5.Average = (Sum of items)/Number of items

Permutations and Combinations :
1.nPr = n!/(n-r)!
2.nPn = n!
3.nP1 = n

1.nCr = n!/(r! (n-r)!)
2.nC1 = n
3.nC0 = 1 = nCn
4.nCr = nCn-r
5.nCr = nPr/r!

Number of diagonals in a geometric figure of n sides = nC2-n

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.

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)

Percentage :
1.If A is R% more than B, then B is less than A by R / (100+R) * 100
2.If A is R% less than B, then B is more than A by R / (100-R) * 100
3.If the price of a commodity increases by R%, then reduction in consumption, not to increase the expenditure is : R/(100+R)*100
4.If the price of a commodity decreases by R%, then the increase in consumption, not to decrease the expenditure is : R/(100-R)*100

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|

Profit & Loss :
1.Gain = Selling Price(S.P.) – Cost Price(C.P)
2.Loss = C.P. – S.P.
3.Gain % = Gain * 100 / C.P.
4.Loss % = Loss * 100 / C.P.
5.S.P. = (100+Gain%)/100*C.P.
6.S.P. = (100-Loss%)/100*C.P.

Time & Work :
1.If A can do a piece of work in n days, then A’s 1 day’s work = 1/n
2.If A and B work together for n days, then (A+B)’s 1 days’s work = 1/n
3.If A is twice as good workman as B, then ratio of work done by A and B = 2:1

Time & Distance :
1.Distance = Speed * Time
2.1 km/hr = 5/18 m/sec
3.1 m/sec = 18/5 km/hr
4.Suppose a man covers a certain distance at x kmph and an equal distance at y kmph. Then, the average speed during the whole journey is 2xy/(x+y) kmph.

Logarithms :
If am = x , then m = logax.
Properties :
1.log xx = 1
2.log x1 = 0
3.log a(xy) = log ax + log ay
4.log a(x/y) = log ax – log ay
5.log ax = 1/log xa
6.log a(xp) = p(log ax)
7.log ax = log bx/log ba
Note : Logarithms for base 1 does not exist.

Simple & Compound Interests :
Let P be the principal, R be the interest rate percent per annum, and N be the time period.
1.Simple Interest = (P*N*R)/100
2.Compound Interest = P(1 + R/100)N – P
3.Amount = Principal + Interest

Problems on Trains :
1.Time taken by a train x metres long in passing a signal post or a pole or a standing man is equal to the time taken by the train to cover x metres.
2.Time taken by a train x metres long in passing a stationary object of length y metres is equal to the time taken by the train to cover x+y metres.
3.Suppose two trains are moving in the same direction at u kmph and v kmph such that u>v, then their relative speed = u-v kmph.
4.If two trains of length x km and y km are moving in the same direction at u kmph and v kmph, where u>v, then time taken by the faster train to cross the slower train = (x+y)/(u-v) hours.
5.Suppose two trains are moving in opposite directions at u kmph and v kmph. Then, their relative speed = (u+v) kmph.
6.If two trains of length x km and y km are moving in the opposite directions at u kmph and v kmph, then time taken by the trains to cross each other = (x+y)/(u+v)hours.
7.If two trains start at the same time from two points A and B towards each other and after crossing they take a and b hours in reaching B and A respectively, then A’s speed : B’s speed = (vb : v 