Some Mathematical Notations and Formulas

1. Exponentials

• For any real number a ≠ 0, m, and n, we have the following identities:
  • a⁰ = 1
  • a¹ = a
  • a^-1 = 1 / a
  • a^-c = 1 / a^c
  • (a^m)ⁿ = a^mn
  • (a^m)ⁿ = (aⁿ)^m
  • a^maⁿ = a^m+n
  • (a ×b)^x = a^x ×b^x 
  • c ×a^x = (p ×a^x) + (q ×a^x), where c = p + q

• Examples:
  • aⁿ⁺² = aⁿ * a²
  • 5*2ⁿ = 4*2ⁿ + 1*2ⁿ = (2²)*2ⁿ + 1*2ⁿ = 2ⁿ⁺² + 2ⁿ

2. Factorials

• The notation n! is defined for integers n >= 0 as

  

• Examples: 

  0! = 1
  1! = 1 * 0! = 1
  2! = 2 * 1! = 2 * 1 * 0! = 2
  3! = 3 * 2! = 3 * 2 * 1! = 3 * 2 * 1 * 0! = 6

3.. Logarithms

• A logarithm of base a

  x = log_ab ≡ a^x = b.

• Some properties of logarithms.
  • log_a(x*y) = log_ax + log_ay 
  • log_a(x^y) = y*log_ax 
  • log_a(x/y) = log_ax - log_ay
  • log_ax = log_bx / log_ba

• Examples:
  • log₂5 = log₁₀5 / log₁₀2
  • k = 2^x Û log₂k = x

4. Summations

• Notation

  Summation of a sequence/series [a₁, a₂,.., a_n] is denoted using a symbol S.
   

• Arithmetic series

  

  • Example with coefficients:
    

• Geometric series

  

  • Examples:

    

    

• Telescoping series

  

5. Quadratic equations