[AM_20150302BTRCUK] Binomial with known Ratio of Coefficients but Unknown Power

Question

Introduction

     This Additional Mathematics question is challenging because the index  n  of the binomial power is unknown but you only know the ratio of a pair of coefficients.  Usually the  n  is given and you just plug in the formula and apply the binomial expansion formula.

Reminder

Solution

   

Heuristics Used

H05. Work backwards

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* GCE ‘A’ Levels H2 Mathematics (revision)

* IB Mathematics (revision)

* other syllabuses that involve Binomial Theorem for higher powers

[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, 3^rd 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.

HeuristicsUsed

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

[JCH2BXQNSR_20150429] Binomial Expansion for a Quotient

Question

Solution

Comment

     Note in the above working that  3(x + kx²)² = 3(x² + 2kx³ + k²x⁴),  but since we do not need the  x³  and  x⁴  terms, we omit them and just write “3(x² + ... )”.

Heuristics Used

H12* Think of a related problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* revision for GCE ‘A’ Level H2 Mathematics

* AP Calculus

* other syllabuses infinite binomial series

[AM_20150409ASD] Absurd Surds!

Question

Introduction

     This looks like a dastardly absurd problem on surds.  If you tried to do this question by direct calculation, you might get the correct answer after a long series of working, provided you do not make careless mistakes.

     Is there a better way to do it?  You bet!  First we simplify  x  as much as possible.  [H10]  Secondly, look for relationships or patterns.  [H04]  Note that  y  is the “inverted” version of  x, so  y  is the reciprocal of  x.  Thirdly, use more algebra [H13] instead of just calculating like a donkey.  Some useful algebraic identities are  x³ + y³ = (x + y)(x² – xy + y²),  and   x² + y² = (x + y)² – 2xy.

Solution

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels
* GCE ‘O’ Level Additional Mathematics
* anyone who is interested in surds, even if it seems absurd.