### NUMBER THEORY

**1.** Sum of natural numbers from 1 to n

e.g Sum of natural numbers from 1 to 40 = 40(40+1)/2 = 820

**2.** Sum of squares of first n natural numbers is =

**3.** Sum of the squares of first n even natural numbers is

**4.** Sum of cubes of first n natural numbers is

**5.** Any number N can be represented in the decimal system of number as

**Important Formulas**

**i.** ( a + b )( a – b ) = ( a ^{2} – b ^{2} )

**ii.** ( a + b ) ^{2} = ( a ^{2} + b ^{2}+ 2 ab )

**iii.** ( a – b )^{2} = ( a^{2} + b ^{2} – 2 ab )

**iv.** ( a + b + c ) ^{2} = a^{2} + b^{2} + c ^{2} + 2 ( ab + bc + ca )

**v.** ( a ^{3} + b^{3} ) = ( a + b )( a ^{2} – ab + b ^{2} )

**vi.** ( a ^{3} – b ^{3} ) = ( a – b )( a ^{2} + ab + b ^{2} )

**vii.** ( a ^{3} + b ^{3} + c ^{3} – 3 abc ) = ( a + b + c )( a ^{2} + b ^{2} + c ^{2} – ab – bc – ac )

**viii.** When a + b + c = 0, then a ^{3} + b ^{3} + c ^{3} = 3 abc .

**xi.** ( a + b ) ^{2} = ( a ^{2} + b ^{2} + 2 ab ) =(a – b)^{2} + 4ab

**x.** ( a – b ) ^{2} = ( a ^{2} + b ^{2} – 2 ab ) = (a + b)^{2} – 4ab

**Some more tips:**

**1)** k(a + b + c) = ka + kb + kc

**2)** (a + b) (c + d) = ac + ad + bc + bd

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

**4)** ( a + b ) ^{2} – ( a – b ) ^{2} = 4ab

**5)** ( a + b ) ^{2} – ( a – b ) ^{2} = 2(a^{2} + b^{2})

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

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

**8)** 1/a + 1/b = (a + b)/ ab

**9)** (x + a)(x + b)(x + c) = x^{3} + (a +b +c)x^{2} + (ab + bc + ca)x +abc

**10)** (a + b + c)^{3} = a^{3} + b^{3} + c^{3} + 3a^{2}b + 3a^{2}c + 3b^{2}a +3b^{2}c + 3c^{2}a + 3c^{2}b + 6abc