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MIRACLE TRIANGLE

 Triangle of numbers called pascal triangle has so many properties.  It touches all the branches of mathematics.  This one article is not enough to describe the glory of the triangle.  So we will meet this miracle maths object now and then.

    Construction:  Start with "1".  Place numbers below it in triangular pattern based on this rule.
" Each number is the sum of the pair of numbers residing above it."(figure 1)
There are so many patterns in the triangle.  We will see some of them.(figure 1)


 The first diagonal is just 1s.  The second diagonal has the counting numbers 1,2,3,etc,.  The third diagonal has the triangular numbers.  The triangle is symmetrical.  The left side of the triangle is the mirror image of the right side.  The sum of each row gives the powers of  2.




     The rows in pascal's triangle can also show you the coefficients of binomial expansion as given figure 2.  The square of the number in the second diagonal is the sum of the numbers next to it and below to it.

      Besides these patterns; prime numbers , Fibonacci sequence, probability, combinations are hidden in the triangle.  Try to explore them and have fun.

   science update; The missing chandrayaan is found to orbit moon by NASA 
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