Basic Formulas :

(a + b)2 = a2 + b2 + 2ab

(a − b)2 = a2 + b2 − 2ab

(a + b)(a − b) = a2 − b2

(a + b)2 − (a − b)2 = 4ab

(a + b)2 + (a − b)2 = 2 (a2 + b2)

(a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ac)

(a3 + b3) = (a + b) (a2 − ab + b2)

(a3 − b3) = (a − b) (a2 + ab + b2)

(a + b)3 = a3 + b3 + 3ab(a + b)

(a − b)3 = a3 − b3 − 3ab(a − b)

(a3 + b3 + c3 − 3abc) = (a + b + c) (a2 + b2 + c2 − (ab + bc + ca))

If a + b + c = 0, then a3 + b3 + c3 = 3abc

Series :

(1 + 2 + 3 + 4 + .... + n) = n (n + 1)

(12 + 22 + 32 + ..... + n2) = 16 n (n + 1) (2n + 1)

(13 + 23 + 33 + ..... + n3) = 14 n2 (n + 1)2

Operations on Odd & Even numbers :

ODDS EVENS ODDS & EVENS
Odd x Odd = Odd Even x Even = Even Odd x Even = Even
Odd + Odd = Odd Even + Even = Even Odd + Even = Odd
Odd − Odd = Odd Even − Even = Even Odd − Even = Odd
Odd ÷ Odd = Odd Even ÷ Even = Even or Odd Even ÷ Odd = Even
Note : Odd ÷ Even → Not divisible

Important Division Results :

(xn − an) is divisible by (x − a) for all values of n.

(xn − an) is divisible by (x + a) for even values of n.

(xn + an) is divisible by (x + a) for odd values of n.