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Mathematics Dictionary
Dr. K. G. Shih
Sequences : Triangular Patterns


  • Q01 | - Numbers in triangular patterns
  • Q02 | - Difference of sequence of 1, 3, 6, 10, 15, ....
  • Q03 | - Find S(n) = 1 + 3 + 6 + 10 + 15 + .....
  • Q04 | - Number pattern : Top row numbers ib triangular sequence
  • Q05 | - Triangles in triangles


Q01. Numbers in square pattern

Sequence : 1, 3, 6, 10, ..... find nth term
    * .................. 1

    *
    * * ................ 3 = T(1) + 2

    *
    * *
    * * * .............. 6 = T(2) + 3

    *
    * *
    * * *
    * * * * ............ 10 = T(3) + 4




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Q02. Squences : 1, 3, 6, 10, 15, 21, ....

Difference
    * 1st difference F(k) = T(k + 1) - T(K) : 2 3 4 5 6 .......
    * 2nd difference G(k) = F(k + 1) - F(k) : 1 1 1 1 1 .......
Formula
    * nth term : T(n) = n*(n+1)
    * Sum of n terms : S(n) = n*(n + 1)*(n + 1)/6
    * This is same as quadratic function y = (x^2)/2 + x/2
    * 2nd derivative y" = 1 and it is the same as 2nd difference

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Q03 Find S(n) = 1 + 3 + 6 + 10 + ..... + n*(n+1)/2

Prove that S(n) = n*(n + 1)*(n + 2)/6
  • Sum[n] = n*(n + 1)/2
  • Sum[n^2] = n*(n + 1)*(2*n + 1)/6
  • Sum[n*(n + 1)/2]
    • = Sum[(n^2)/2] + Sum[n/2]
    • = n*(n + 1)*(2*n + 1)/12 + n*(n + 1)/4
    • = n*(n + 1)*((2*n + 1)/12 + 1/4)
    • = n*(n + 1)*(n + 2)/6

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Q04 Number pattern : Top row numbers ib triangular sequence

Number pattern (1)
  • Top row Numbers in triangular sequence
  • Example
    • 1. Find row number and column number for 100
    • 2. What is the number at column 6 and row 6 ?

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Q05. Triangles in triangles Pattern

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