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Mathematics Dictionary
Dr. K. G. Shih
Senior High Mathematics
Z041 : Determinant of order 4



Questions
  • Q01 | - Pascal triangle
  • Q02 | - Sequence at r1 : 1, 2, 3, 4, 5, ....
  • Q03 | - Sequence at r2 : 1, 3, 6, 10, 15, 21, .....
  • Q04 | - Prove that Sum[n*(n + 1)/2] = n*(n + 1)*(n + 2)/6
  • Q05 | - Sequence : Prove that Sum[C(n + 1, 2)] = C(n + 2, 3)


Q01. Pascal triangle

  • n .... r0 .... r1 .... r2 .... r3 .... r5 .... r6 .... r7 .... r8
  • 0 ..... 1
  • 1 ..... 1 ..... 1
  • 2 ..... 1 ..... 2 ..... 1
  • 3 ..... 1 ..... 3 ..... 3 ..... 1
  • 4 ..... 1 ..... 4 ..... 6 ..... 4 ..... 1
  • 5 ..... 1 ..... 5 .... 10 .... 10 ..... 5 ..... 1
  • 6 ..... 1 ..... 6 .... 15 .... 20 .... 15 ..... 6 ..... 1
  • 7 ..... 1 ..... 7 .... 21 .... 35 .... 35 .... 21 ..... 7 ..... 1
Example : Expand (x + y)^6
  • (x + y)^5 = x^5 + 5*(x^4)*y + 10*(x^3)*(y^2) + 10*(x^2)*(y^2) + 5*x*(y^4) + y^5
  • The coefficients are in Pascal triangle at n = 5

Go to Begin

2. Sequence at r1

Sequence
  • 1, 2, 3, 4, 5, 6, ....
  • T(n) = nth term = n
  • S(n) = Sum of n terms = n*(n + 1)/2

Go to Begin

Q03.Sequence at r2

Sequence
  • 1, 3, 6, 10, 15, 21, ....
  • T(n) = nth term = n*(n + 1)/2
  • S(n) = Sum of n terms = n*(n + 1)*(n + 2)/6

Go to Begin

Q04. Prove that Sum[n*(n + 1)/2] = n*(n + 1)*(n + 2)/6

C(n, r) is coeff of binomial expansion.
  • 1. What is Pascal triangle ?
  • 2. Prove that Sum[n*(n + 1)/2] = n*(n + 1)*(n + 2)/3!
  • 3. Prove that Sum[C(n + 1, 2)] = C(n + 2, 3)
Reference
Go to Begin

Q05. Sequence from C(n+1, 2)

Questions
  • 1. Write down the sequence
  • 2. Find the nth term T(n)
  • 3. Find the sum S(n)
Solutions
  • 1. The sequence
    • n = 1 : C(2, 2) = 1
    • n = 2 : C(3, 2) = 3*2/(2!) = 3
    • n = 3 : C(4, 2) = 4*3/(2!) = 6
    • n = 4 : C(5, 2) = 5*4/(2!) = 10
    • n = 5 : C(6, 2) = 6*5/(2!) = 15
    • Etc.
    • This sequence is the triangular number patterns
    • It is 1, 3, 6, 10, 15, 21, .... as shown in Pascal triangle at r2 column
    2. T(n)
    • T(n) = C(n + 1, 2) = n*(n + 1)/(2!)
    3. S(n)
    • S(n) = Sum[C(n + 1), 2] = C(n + 2, 3)
    • S(n) = Sum[n*(n + 1)/2] = n*(n + 1)*(n + 2)/(3!)
Pscal triangle with r
    Column ...... r0 r1 r2 r3 r4 r5 r6 r7 r8
    ---------------------------------
    n=0 ......... 01
    n=1 ......... 01 01
    n=2 ......... 01 02 01
    n=3 ......... 01 03 03 01
    n=4 ......... 01 04 06 04 01
    n=5 ......... 01 05 10 10 05 01
    n=6 ......... 01 06 15 20 15 06 01
    n=7 ......... 01 07 21 35 35 21 07 01
    n=8 ......... 01 08 28 56 70 56 28 08 01
Go to Begin
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