Pascal’s Triangle - “Triangular Numbers Squares”The triangular numbers 1, 3, 6, 10,... appear in the second diagonal of Pascal’s Triangle.Take the sum of two consecutive triangular numbers and notice that you get a perfect square: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, etc.Provide an argument as to why this happens:An algebraic argument:We can write the triangular numbers using binomial coefficients (“choose” #’s) as 4C2 + 5C2 = 16. i) Using the formula nC2 = n(n-1)/2, verify that 4C2 + 5C2 = 16.ii) How does this generalize? Rewrite the above example using the variable n, then provide an algebraic proof that shows it is true for all values of n.

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Answered: Pascal’s Triangle - “Triangular Numbers…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Pascal’s Triangle - “Triangular Numbers Squares”

The triangular numbers 1, 3, 6, 10,... appear in the second diagonal of Pascal’s Triangle.

Take the sum of two consecutive triangular numbers and notice that you get a perfect square: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, etc.

Provide an argument as to why this happens:

An algebraic argument:

We can write the triangular numbers using binomial coefficients (“choose” #’s)

as 4C2 + 5C2 = 16.

i) Using the formula nC2 = n(n-1)/2, verify that 4C2 + 5C2 = 16.
ii) How does this generalize? Rewrite the above example using the variable n, then provide an algebraic proof that shows it is true for all values of n.

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