18M.2.AHL.TZ2.H_5

pestleMathematicsAAHLPaper 218M· sl-1-9-binomial-theorem-where-n-is-an-integersource ↗

Express the binomial coefficient $\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right)$ as a polynomial in $n$ .

[3]

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

[3]

Markscheme / solution

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

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

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

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

[3 marks]

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

$n > 60.57 \ldots$ (A1)

Note: Allow equality.

$\Rightarrow n = 61$ A1

[3 marks]

Examiners’ report

[N/A]