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Mathematics Dictionary
Dr. K. G. Shih
Binomial expansion coefficients


  • Q01 | - Prove that 5^(2*n) - 24*n - 1 is divisible by 576
  • Q02 | - Prove that 3^n - 2*n - 1 is divisible by 4
  • Q03 | - Prove that 3^(2*n) - 8*n - 1 is divisible by 64
  • Q04 | - Prove that 2^(3*n) - 7*n - 1 is divisible by 49

    • Q01. Prove that 5^(2*n) - 24*n - 1 is divisible by 576

    Binomial Theorem
    • (x + y)^n = C(n,0)*x^n + C(n,1)*(x^(n-1))*(y)+ C(n,2)*(x^(n-2))*(y^2) + ....
    • Coefficients of expansion
      • C(n,r) = n*(n-1)*(n-2)*...*(n-r+1)/r!.
      • C(n,0) = Coefficient of first term = 1.
      • C(n,n) = Coefficient of last term = 1
      • C(n,1) = Coefficient of 2nd term = n
      • C(n,n-1) = Coefficient of last 2nd term = n
    • It can be expressed as (x + y)^n = Sum[C(n,r)*(x^(n-r))*(y^r)]
    Keyword and question
    • Binomial expansion coefficients
    Proof
    • Expression
      • 5^(2*n) - 24*n - 1
      • = (5^2)^n - 24*n - 1
      • = (1 + 24)^n - 24*n - 1
      • = C(n,0) + C(n,1)*24 + C(n,2)*(24^2) + ...... + C(n,n)*(24^n) - 24*n - 1
    • Since c(n,0) = 1 and C(n,1) = n and C(n,r) is integers
    • Hence 5^(2*n) - 24*n - 1
      • = C(n,2)*(24^2) + C(n,3)*(24^3) + ...... + C(n,n)*(24^n)
      • = (24^2)*(C(n,2) + C(n,3)*(24^1) + ..... + C(n,n)*(24^(n-2))
    • Since (C(n,2) + C(n,3)*(24^1) + ..... + C(n,n)*(24^(n-2)) is integer
    • Hence 5^(2*n) - 24*n - 1 is divisible by 24^2 or 576

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    Q02. Prove that 3^n - 2*n - 1 is divisible by 4

    Proof
    • Expression
      • = 3^n - 2*n - 1
      • = (2 + 1)^n - 2*n - 1
      • = 2^n + C(n,1)*2^(n-1) + C(n,2)*2^(n-2) + ... C(n,n-1)*2 + C(n,n) - 2*n - 1
    • Since C(n,n) = 1
    • Since C(n, n-1) = n
    • Hence Expression
      • = 2^n + C(n,1)*2^(n-1) + .... C(n,n-2)*2^2
      • = 4*(2^(n-2) + C(n,1)*2^(n-3) + ..... C(n,n-2))
    • Since Coefficients C(n,r) are integers
    • Hence it is divisible by 4
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    Q03. Prove that 3^(2*n) - 8*n - 1 is divisible by 64

    Proof
    • Expression
      • = 3^(2*n) - 8*n - 1
      • = (8 + 1)^n - 8*n - 1
      • = 8^n + C(n,1)*8^(n-1) + ... + C(n,n-1)*8 + C(n,n) - 8*n - 1
    • Since C(n,n-1) = n and C(n,n) = 1
    • Expression
      • = 8^n + C(n,1)*8^(n-1) + ... + C(n,n-2)*8^2
      • = (8^2)*(8^(n-2) + C(n,1)*(8^(n-3) + .... C(n,n-2))
    • Since coefficients C(n,r) are positive
    • The expression is divisble by 64

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    Q4. Prove that 2^(3*n) - 7*n - 1 is divisible by 49

    Proof
    • Expression
      • = 2^(3*n) - 7*n - 1
      • = (7 + 1)^n - 7*n - 1
      • = 7^n + C(n,1)*7^(n-1) + ... + C(n,n-1)*7 + C(n,n) - 7*n - 1
    • Since C(n,n-1) = n and C(n,n) = 1
    • Expression
      • = 7^n + C(n,1)*7^(n-1) + ... + C(n,n-2)*7^2
      • = (7^2)*(7^(n-2) + C(n,1)*(7^(n-3) + .... C(n,n-2))
    • Since coefficients C(n,r) are positive
    • The expression is divisble by 49

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