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


  • Q01 | - Numbers in square patterns
  • Q02 | - Difference of sequence of 1, 4, 9, 16, 25, ....
  • Q03 | - Find S(n) = 1 + 4 + 9 + 16 + ..... + n^2
  • Q04 | - Squares in squares


Q01. Numbers in square pattern

Sequence : 1, 3, 6, 10, ..... find nth term
    * .................. 1^2

    * *
    * * ................ 2^2

    * * *
    * * *
    * * * .............. 3^2

    * * * *
    * * * *
    * * * *
    * * * * ............ 4^2


    Hence : T(n) = n^2
    Hence : S(n) = n*(n + 1)*(2*n + 1)/6


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Q02. Squences : 1, 4, 9, 16, 25, 36, ....

Difference
    * 1st difference F(k) = T(k + 1) - T(K) : 3 5 7 9 11 ......
    * 2nd difference G(k) = F(k + 1) - F(k) : 2 2 2 2 2 .......
Formula
    * nth term : T(n) = n^2
    * Sum of n terms : S(n) = n*(n + 1)*(2*n + 2)/6
    * This is same as quadratic function y = x^2
    * 2nd derivative y" = 2 and it is the same as 2nd difference

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Q03 Find S(n) = 1 + 4 + 9 + 16 + ..... + n^2

Prove that S(n) = n*(n + 1)*(2*n + 1)/6
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Q04 Squares in squares patterns

Squres in squares patterns Examples
  • 1. Sketch pattern for 4^2
  • 2. Sketch pattern for 5^2

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Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
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