Mathematics Dictionary

Mathematics Dictionary
Dr. K. G. Shih

Tetrahedral Numbers

Questions

Symbol Defintion

Sqr(x) = Square root of x

Q01 | - Pentatope number in Pascal triangle

Q02 | - Pentatope number : Patterns

Q03 | - Pentatope number : Prove T(n) = n*(n+1)*(n+2)/6

Q04 | - Pentatope number : Prove S(n) = n*(n+1)*(n+2)*(n+3)/4!

Q05 | - Pentatope number : Prove that Sum[C(n+2),3)] = C(n+3,4)

Q06 | - Using Sum[n*(n+1)*(n+2)*(n+3)/24] find Sum[n^4]

Q07 | - Pentatope number : Pattern

Q08 | -

Q09 | -

Q10 | -

Answers

Q01. Tetrahedral number in Pascal triangle
Pascal triangle

      C1 C2 C3 C4 C5 C6 C7 C8 ....

  R1  1
  R2  1  1
  R3  1  2  1
  R4  1  3  3  1
  R5  1  4  6  4  1
  R6  1  5 10 10  5  1
  R7  1  6 15 20 15  6  1
  R8  1  7 21 35 35 21  7  1
  R9  1  8 28 56 70 56 28  8  1

Pentatope number

 1. Pentatope numbers in column 5 (C5)
 2. The numbers : 1, 5, 15, 35, 70, 126, 210, ... 
 2. 1st Diff    : 1, 4, 10, 20, 35,  56,  84, ...
 3. 2nd Diff    : 1, 3,  6, 10, 15,  21,  28, ...
 4. 3rd Diff    : 1, 2,  3,  4,  5,   6,   7, ...
 5. 4th Diff    : 1, 1,  1,  1,  1,   1,   1, ...

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Q02. Find Pentatope number sequence using Tetrahedral numbers
Method

  Pentatope  sequence : 1   5  15
  Tetraheral sequence : 1   4  10   20  35   56   84 .....

  2nd number of tetrahedral number is  1 +  4 =   5
  3rd number of tetrahedral number is  5 + 10 =  15
  4th number of tetrahedral number is 15 + 20 =  35
  5th number of tetrahedral number is 35 + 35 =  70
  2nd number of tetrahedral number is 70 + 56 = 126
                                       |    |     |
                                       |    |     Pentatope Seq 
                                       |    Tetrahedral numbers
                                       Pentatope seq

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Q03. Pentatope number : T(n) = n*(n+1)*(n+2)*(n+3)/24

Prove nth term is T(n) = n*(n+1)*(n+2)*(n+3)/24

   T(1) =  1
   T(2) =  5 = 1 + 4
   T(3) = 15 = 1 + 4 + 10
   T(4) = 35 = 1 + 4 + 10 + 20
   T(5) = 70 = 1 + 4 + 10 + 20 + 35 (Sum of tetrahedral numbers)
   ....
   T(n) = Sum[n*(n+1)*(n+2)/6] = n*(n+1)*(n+2)*(n+3)/24

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Q04. Pentatope number : Sum[T(n)] = n*(n+1)*(n+2)*(n+3)*(n+4)/5!

Proof

   S(n)   = n*(n+1) ........................... (1)
   S(n^2) = n*(n+1)*(2*n+1)/6 ................. (2)
   S(n^3) = (n*(n+1)/2)^2 ..................... (3)
   S(n^4) =                                     (4)
   Sum[T(n)] = Sum[n*(n+1)*(n+2)*(n+3)/24]
             = ((Sum[n^4 + ...)] .............. (5)
   Substitute (1), (2), (3), (4) into (5) and simplify we have
   Sum[T(n)] = Sum[n*(n+1)*(n+2)*(n+3)/24]
             = n*(n+1)*(n+2)*(n+3)*(n+4)/5!

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Q05. Pentatope : Sum[C(n+3,4)] = C(n+4,5)

   1. Binomial expansion coefficients
      C(m,r) = m*(m-1)*(m-2) .... (m-r+1)/r!
   2. Hence C(n+3,4) = (n+3)*(n+2)*(n+1)*n/4!
            C(n+4,5) = (n+4)*(n+3)*(n+2)*(n+1)*n/5!
   3. Hence from Q04 we have Sum[C(n+3,4)] = C(n+4,5)

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Q06. Using Sum[n*(n+1)*(n+2)*(n+3)/4!] find Sum[n^4]

Proof : We need following formula

  Sum[n]   = n*(n+1)/2
  Sum[n^2] = n*(n+1)*(n+2)/6 
  Sum[n^3] =
  Sum[n*(n+1)*(n+2)*(n+3)/24] = n*(n+1)*(n+2)*(n+3)*(n+4)/5!

Reference
  More methods to prove this formula are given in keyword series
  about Sum[n^4] in www.b192907.com




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Q07. Tetrahedral number : Pattern


The pattern in tetrhedron (4 surfaces, each is triangle)
    *    *     * ----- 1 = top          * --------- 1 = top

         *     *                        *
        * *   * * ---- 3 = middle      * * -------- 2nd layer

               *                        *
              * *                      * *
             * * * --- 6 = bottom     * * * ------- 3rd layer

                                        *
                                       * *
                                      * * *
                                     * * * * ------ 4th layer = base

    1  4 = 1+3  10 = 1+3+6             20 = 1+3+6+10

   




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Q08. Answer







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Q09. Answer







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Q10. Answer







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