Important formulas for cds exam 2022 (1)
Exam Target
(a+b)2 = a2 + b2 + 2ab
(a-b)2 = a2 -2ab + b2
(a + b)3 = a3 + b3 + 3ab(a+b)
(a - b)3 = a3 – b3 - 3a2b+ 3ab2
a2 – b2 = (a-b)(a+b)
(a+b)2 – (a-b)2 = 2(a2 + b2)
(a+b)2 – (a-b)2 = 4ab
(a+b)3 – (a-b)3 =2a(a2 +3b2)
(a+b)3 – (a-b)3 = 2b(3a2+b2)
a3+b3+c3-3ab = (a+b+c)[a2+b2+c2-2ab-2ac-2bc] = (a+b+c)/2 [(a-b)2 + (b-c)2 + (c-a)2 ]
= (a+b+c)[ (a+b+c)2 - 3(ab+bc+ca)]
a2 + b2 + c2 - ab – bc – ca = 1/2 [ (a-b)2 + (b-c)2 + (c-a)2 ]
if a+b+c=0
then a3 + b3 + c3 = 3abc
(14) sum of first n natural numbers is
Remainder theorem important formula:
n(n+1)/2
(15) sum of first n even number is = n(n+1)
(16) sum of first odd number n2
(17) first n natural number square sum
1/6n[(2n+1) (n+1) ]
(18) first n sum number square sum is
2/3 n(n+1)(2n+1)
(19) first n odd number square sum is
n/3 [4n2 -1]
(20) first n natural number cube sum
[n/2(n+1)]2
(21) first n odd number cube sum
n2(2n2-1)
(22) first n even number cube sum
2n2(n+1)2
