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G - Pascal's Triangle Investigation

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					                     Pascal’s Triangle & Binomial Expansion

                       The accompanying sheet has an arrangement of numbers called
                     ‘Pascal’s Triangle’. It is named after a French mathematician, Blaise
                       Pascal (1623 – 1662), who discovered this unique arrangement of
                    numbers at the age of 13. It contains several interesting relationships
                           among its numbers, with some of them being recursive.


  1. What patterns do you see in the arrangement of numbers that have been filled in?
  2. Can you predict the next row of numbers?
  3. Do any numbers repeat?
  4. Complete the rest of the triangle, using the patterns discovered above.


Onto some cool stuff…
  5. Do you see a pattern in the sums of the numbers in each row?
  6. See if you can find a pattern of natural numbers (1, 2, 3, 4, …)
  7. See if you can find a pattern of ‘triangular numbers’
  8. See if you can find ‘Fibonacci numbers’


        Fibonacci numbers follow                     Triangular numbers follow
       the pattern 1, 1, 2, 3, 5, 8,…,              the pattern 1, 3, 6, 10,… (the
        where each term is the sum                    number of dots needed to
            of the previous two.                          make a triangle)



Even more cool stuff…

   In addition to triangular numbers, there are
  other polygonal numbers, such as square,
  pentagonal, hexagonal, etc… Looking at the
  number patterns below (and the diagrams to
 help understand these patterns), see if you can
         find these in Pascal’s Triangle.

   square numbers  1, 4, 9, 16, 25, 36, 49, …
 pentagonal numbers  1, 5, 12, 22, 35, 51, 70, …
 hexagonal numbers  1, 6, 15, 28, 45, 66, 91, …
  9. To investigate how Pascal’s Triangle can be used in binomial expansion, study the
     following table and answer the questions below

Factored Form                                         Expanded Form

    a  b 1                        a b

    a  b 2                   a 2  2ab  b 2

    a  b 3               a 3  3a 2b  3ab2  b3

    a  b 4          a 4  4a3b  6a 2b 2  4ab3  b 4

    a  b 5      a5  5a 4b  10a3b 2  10a 2b3  5ab4  b5

  10. Describe how the degree of each term relates to the exponent of the binomial.
  11. Compare the numbers in Pascal’s Triangle to your binomial expansions and describe
      any pattern you find.
  12. Use the information from #’s 10 & 11 to predict the expansion of a  b  .
                                                                               6




                Any binomial can be expanded using Pascal’s Triangle to
                      help determine the coefficients of each term.

Now give these a try…

  13. Expand and simplify  x  3
                                      4




  14. Expand and simplify 5a  2b 
                                          8




                            
  15. Expand and simplify 2 x  3 y
                                 3
                                              
                                            2 5

				
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posted:12/4/2011
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