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# Mathematics-Equation Corner

**Laws of Indices::-**

(1)

(1)

**a**

^{m}x a^{n}= a

^{m + n}(2) a^{m}/ a^{n}= a

(3) (a

^{m - n}(3) (a

^{m})^{n}= a

(4) (ab)

^{mn}(4) (ab)

^{n}= a^{n}b

(5) (a / b)

(6) a

^{n}(5) (a / b)

^{n}= a^{n}/ b^{n}(6) a

^{0}= 1**Laws of Surds ::-**

(1)

(1)

**a**

(2)

(3) /

^{1/3}(2)

(3) /

**Sum of n Terms ::-**

**(1 + 2 + 3 + . . . . + n ) = n ( n + 1 ) / 2**

**Sum of squares n Terms ::-**

( 1

( 1

^{2}+ 2^{2}+ . . . . . .+ n^{2}) = n (n + 1 ) (2n + 1) / 6**Sum of cube n Terms ::-**

**( 1**

Arithmetic Progression ( A.P ) :: -

^{3}+^{ }2^{3}+ 3^{3}+ . . . . .+ n^{3}) = n^{2}( n + 1 )^{2}/ 4Arithmetic Progression ( A.P ) :: -

**a, a + d, a + 2d, . . . are said to be in A.P, in which a = first term, d = common difference**

Then n

Sum of n terms = n [ 2a + ( n - 1 ) d ]/ 2

Geometric Progression ( G.P ) ::-

Then n

^{th}term of A.P = a + ( n - 1 ) dSum of n terms = n [ 2a + ( n - 1 ) d ]/ 2

Geometric Progression ( G.P ) ::-

**a, ar**

So n

= a ( r

^{2}, ar^{3}, ar^{4}, . . . . . are said to be G.P, in which a = first term and r = common ratio.So n

^{th}term = a ( 1 - r^{n}) / ( 1 - r ) if r < 1= a ( r

^{n}- 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

L.C.M ::-

L.C.M of Numerators / H.C.F of Denominators

**Basic Formulae :: -**

**( 1 )**

**( a + b ) ( a - b ) = ( a**

( 2 ) ( a + b )

( 3 ) ( a - b )

( 4 ) ( a + b + c )

( 5 ) ( a

( 6 ) ( a

( 7 ) ( a

( 8 ) If ( a + b + c ) = 0, then ( a

^{2}- b^{2})( 2 ) ( a + b )

^{2}= ( a^{2}+ b^{2}+ 2ab )( 3 ) ( a - b )

^{2}= ( a^{2}+ b^{2}- 2ab )( 4 ) ( a + b + c )

^{2}= ( a^{2}+ b^{2}+ c^{2}+ 2ab + 2bc + 2ac )( 5 ) ( a

^{3}+ b^{3}) = ( a + b ) ( a^{2}- ab + b^{2})( 6 ) ( a

^{3}- b^{3}) = ( a - b ) ( a^{2}+ ab + b^{2})( 7 ) ( a

^{3}+ b^{3}+ c^{3}- 3abc ) = ( a + b + c ) ( a^{2}+ b^{2}+ c^{2}- ab - bc - ac )( 8 ) If ( a + b + c ) = 0, then ( a

^{3}+ b^{3}+ c^{3}) = 3abc**Average ::-**

**Sum of observations / Number of observations**

Calculation of Population ::-

Calculation of Population ::-

**Population after n years = p ( 1 + R / 100 )**

Population n years ago = p / ( 1 + R / 100 )

^{n }<=> p ->population of a town, increase at the rate of R% per annumPopulation n years ago = p / ( 1 + R / 100 )

^{n }^{ }**Results on Depreciation :: -**

**Value of machine**

**after n years = p ( 1 - R / 100 )**

Value of machine n years ago = 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 )

( 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 ) = ( a

( 3 ) Sub - Duplicate ratio of ( a : b ) = (√a : √b )

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 ) = ( a

^{2}: b^{2})( 3 ) Sub - Duplicate ratio of ( a : b ) = (

**( 4 )**

**Triplicate ratio of ( a : b ) = ( a**

( 5 ) Sub - Triplicate ratio of ( a : b ) = ( a

( 6 ) Componendo and Dividendo :: - If ( a / b ) = ( c / d ), then ( a + b ) / ( a - b ) = ( c + d ) / ( c - d )

^{3}: b^{3})( 5 ) Sub - Triplicate ratio of ( a : b ) = ( a

^{1/3}: b^{1/3})( 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

( 8 ) If x is inversely proportional to y, then we can write it as xy = k

**( 9 )**

**Mean Proportional between a and b is**