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Mathematics-Equation Corner
Laws of Indices::-
(1) am x an = am + n
(2) am / an = am - n
(3) (am)n = amn
(4) (ab)n = anbn
(5) (a / b)n = an / bn
(6) a0 = 1
Laws of Surds ::-
(1)
a1/3 (2)
(3)
/ 
Sum of n Terms ::-
(1 + 2 + 3 + . . . . + n ) = n ( n + 1 ) / 2
Sum of squares n Terms ::-
( 12 + 22 + . . . . . .+ n2 ) = n (n + 1 ) (2n + 1) / 6
Sum of cube n Terms ::-
( 13 + 23 + 33 + . . . . .+ n3 ) = n2 ( n + 1 )2 / 4
Arithmetic Progression ( A.P ) :: -
a, a + d, a + 2d, . . . are said to be in A.P, in which a = first term, d = common difference
Then nth term of A.P = a + ( n - 1 ) d
Sum of n terms = n [ 2a + ( n - 1 ) d ]/ 2
Geometric Progression ( G.P ) ::-
a, ar2, ar3, ar4 , . . . . . are said to be G.P, in which a = first term and r = common ratio.
So nth term = a ( 1 - rn ) / ( 1 - r ) if r < 1
= a ( rn - 1 ) / ( r - 1 ) if r > 1
H.C.F :: -
H. C. F of Numerators / L.C.M of Denominators
L.C.M ::-
L.C.M of Numerators / H.C.F of Denominators
Basic Formulae :: -
( 1 ) ( a + b ) ( a - b ) = ( a2 - b2 )
( 2 ) ( a + b )2 = ( a2 + b2 + 2ab )
( 3 ) ( a - b )2 = ( a2 + b2 - 2ab )
( 4 ) ( a + b + c )2 = ( a2 + b2 + c2 + 2ab + 2bc + 2ac )
( 5 ) ( a3 + b3 ) = ( a + b ) ( a2 - ab + b2 )
( 6 ) ( a3 - b3 ) = ( a - b ) ( a2 + ab + b2 )
( 7 ) ( a3 + b3 + c3 - 3abc ) = ( a + b + c ) ( a2 + b2 + c2 - ab - bc - ac )
( 8 ) If ( a + b + c ) = 0, then ( a3 + b3 + c3 ) = 3abc
Average ::-
Sum of observations / Number of observations
Calculation of Population ::-
Population after n years = p ( 1 + R / 100 )n <=> p ->population of a town, increase at the rate of R% per annum
Population n years ago = p / ( 1 + R / 100 )n
Results on Depreciation :: -
Value of machine after n years = p ( 1 - R / 100 )n
Value of machine n years ago = p / ( 1 - R / 100 )n
Profit and Loss ::-
( 1 ) Cost Price ::- The price at which the product is purchased, is called cost price, abbreviated as C.P
( 2 ) Selling Price ::- The price at which the product is sold, is called selling price, abbreviated as S.P
( 3 ) Profit or Gain :: - If Selling price is greater than Cost price, then seller is said to have a profit or gain
( 4 ) Loss :: - if Selling price is less than Cost price, then seller is said to have loss
( 5 ) Gain <=> S.P - C.P
( 6 ) Loss <=> C.P - S.P
( 7 ) Gain% = ( Gain * 100 ) / C.P
( 8 ) Loss% = ( Loss * 100 ) / C.P
( 9 ) S.P = ( 100 + Gain% ) * C.P / 100 <=> ( 100 - Loss% ) * C.P / 100
( 10 ) C.P = ( 100 / ( 100 + Gain% ) ) * S.P <=> ( 100 / ( 100 - Loss% ) ) * S.P
( 11 ) If the product is sold at a gain of 40%, then we can say that S. P = 140% of C.P
( 12 ) If the product is sold at a loss of 15%, then we can say that S.P = 85% of C.P
Ratio and Proportion ::-
( 1 ) If a : b = c : d, then we can write a : b :: c : d, and we can say that a, b, c, d are in proportion
Here a, d are called extrems and b, c are called mean terms
Product of extrems = Product of means
( a * d ) = ( b * c )
( 2 ) Duplicate ratio of ( a : b ) = ( a2 : b2 )
( 3 ) Sub - Duplicate ratio of ( a : b ) = ( √a : √b )
( 4 ) Triplicate ratio of ( a : b ) = ( a3 : b3 )
( 5 ) Sub - Triplicate ratio of ( a : b ) = ( a1/3 : b1/3 )
( 6 ) Componendo and Dividendo :: - If ( a / b ) = ( c / d ), then ( a + b ) / ( a - b ) = ( c + d ) / ( c - d )
( 7 ) If x proportional to y, then we can write as x = ky, k is any constant
( 8 ) If x is inversely proportional to y, then we can write it as xy = k
( 9 ) Mean Proportional between a and b is
