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Mathematics Dictionary
Dr. K. G. Shih
Question and Answer
Questions



  • Bible numbers
  • Binary number system
  • Binomial distribution 17 08 : B(x) = C(n,x)*(p^x)*(q^n-x))
  • Binomial expansion
    • 07 02 Binomial expansion : Coeff of (r+1)th term = C(n,r)
    • 03 13 Binomila expansion : Prove that 5^(2*n) - 24*n - 1 is divisible by 576
    • 07 04 Sum[C(n+1),2] = C(n+2,3)
    • 07 05 Sum[C(n+2),3] = C(n+3,4)
    • 07 06 Sum[C(n+3),4] = C(n+4,5)
    • 07 04 Sum[C(n+4),5] = C(n+5,6)
  • Keyword | Binomial expension : Sequences from coefficients
    • 1. Write down the sequence from C(n + 1, 2)
    • 2. Prove that Sum[C(n+1,2)] = C(n+2,3)
    • 3. Prove that Sum[C(n+2,3)] = C(n+3,4)
  • Keyword | Binomial expansion
    • 1. Area under curve y = 1/(1 + x^2)
    • 2. Series of arctan(x)
  • Binomial expansion coefficients
    • Sum[C(n,r)] = 2^n for r = 0, 1, 2, ..... n
    • C(n,0) + C(n,2) + ... = C(n,1) + C(n,3) + .... for n is odd
    • Prove C(n,r) = C(n,n-r)
  • Binomial expansion coefficients
    • 1. Prove that (3^(2*n) - 8*n - 1) is divisible by 64
    • 2. Prove that (2^(3*n) - 7*n - 1) is divisible by 49
  • Binomial expansion coefficients Pascal triangle and series
    • Sum[n] = n*(n+1)/2 = C(n+1,2)
    • Sum[n*(n+1)/2] = n*(n+1)*(n+2)/3! = C(n+2,3)
  • Binomial theory
    • 07 01 T(n) = C(n,r)*(x^(n-r))*(y^r)
    • 07 08 Find Sum[n^4] usimg Sum[C(n+3,4)] = C(n+4,5)
    • 07 09 Fibonacci's sequence in Pascal triangle
    • 07 10 Series from C(n,r)
    • 07 11 Constant oefficient in (x + 1/(x^2))^n
    • 07 12 Expand Sqr(1 + x^2) in series form
    • 07 13 Use Pascal triangle find coefficients of (x+y)^7
    • 07 14 Expand (x+1)^n
    • 07 15 Find coefficient (x^3)*(y^5) in expansion of (x+y)^n
    • 07 16 Expand 1/(1 + x^2)
    • 07 17 C(n,r-1) + C(n,r) = C(n+1,r)
    • 07 18 Coeff of x^(r-1), x^r and x^(r+1) are 3 consecutive AP
    • 07 19 Coefficients of (x^p)*(y^q)*(z^r) in (x+y+z)^n
  • Binomial Theorem : Coefficient C(n,r) and Pascal triangle
  • Keyword | Binomial Theorem : Aplication in Calculus
  • Keyword | Bit and Byte in computer
  • Keyword | Blanking a polygon
  • Keyword | Butterfly Theorem
  • Keyword | Byte and numbers in computer


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