Solved: Express the binomial coefficient ${\binom{3k+1}{3k-2}}$ as a polynomial in... [IB Mathematics Analysis and Approaches (AA)]

Question

HLPaper 2

Express the binomial coefficient ${\binom{3k+1}{3k-2}}$ as a polynomial in ${k}$ .

[3]

Verified

Solution

This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\left( \begin{gathered} 3k + 1 \\\ 3k - 2 \end{gathered} \right) = \frac{{\left( {3k + 1} \right)!}}{{\left( {3k - 2} \right)!3!}}$ (M1)

$= \frac{{\left( {3k + 1} \right)3k\left( {3k - 1} \right)}}{{3!}}$ A1

$= \frac{9}{2}{k^3} - \frac{1}{2}k$ or equivalent A1

[3 marks]

Hence find the least value of $k$ for which $\left( \begin{gathered} 3k + 1 \\\ 3k - 2 \\\ \end{gathered} \right) > {10^6}$ .

[3]

Verified

Solution

attempt to solve $= \frac{9}{2}{k^3} - \frac{1}{2}k > {10^6}$ (M1)

$k > 60.57 \ldots$ (A1)

Note: Allow equality.

$\Rightarrow k = 61$ A1

[3 marks]

Still stuck?

Get step-by-step solutions with Jojo AI

Want more practice questions for Mathematics Analysis and Approaches (AA)?

Footer

RevisionDojo was developed independently of the IBO and as such is not endorsed by it in any way.

SAT® is a trademark registered and owned by the College Board®, which is not affiliated with and does not endorse this product or site.