Thursday, May 21, 2015

[IB-HL H&H_8G Q16] Sum of Squares of some Binomial Coefficients

Question

Introduction
This problem is taken from the Haese textbook for International Baccalaureate, 3rd Edition, page 262.  It looks pretty daunting doesn’t it?  Where do we even begin?  The key to solving this problem is to realise that the binomial coefficients are coefficients of (numbers attached to) certain powers of  x  in the expansion.  The question is:  which power or powers?
Before we go into that, let us review some important relevant facts.

Reminders
Solution

Final Remarks
This problem was solved by using the symmetry property and treating binomial coefficients as coefficients of certain powers of  x.  We also worked backwards by noting that the RHS of the equation to be proven is the coefficient of  xn.  This suggests that we compare this with the coefficients of  xn  on the LHS.

H03. Make a systematic list
H04. Look for pattern(s)
H05. Work backwards
H09. Restate the problem in another way
H13* Use Equation / write a Mathematical Sentence

Suitable Levels
International Baccalaureate Mathematics (HL)
GCE ‘A’ Levels H2 Mathematics
* other syllabuses that involve complex numbers and polynomials