Find the term in ${x^6}$ in the expansion of ${(x + 2)^9}$.

Hence, find the term in ${x^7}$ in the expansion of $5x{(x + 2)^9}$.

valid approach to find the required term     (M1)

eg$\,\,\,\,\,$$\left( {\begin{array}{*{20}{c}} 9 \\ r \end{array}} \right){(x)^{9 - r}}{(2)^r},{\text{ }}{x^9} + 9{x^8}(2) + \left( {\begin{array}{*{20}{c}} 9 \\ 2 \end{array}} \right){x^7}{(2)^2} + \ldots$, Pascal’s triangle to the 9th row

identifying correct term (may be indicated in expansion)     (A1)

eg$\,\,\,\,\,$4th term, $r = 6,{\text{ }}\left( {\begin{array}{*{20}{c}} 9 \\ 3 \end{array}} \right),{\text{ }}{(x)^6}{(2)^3}$

correct calculation (may be seen in expansion)     (A1)

eg$\,\,\,\,\,$$\left( {\begin{array}{*{20}{c}} 9 \\ 3 \end{array}} \right){(x)^6}{(2)^3},{\text{ }}84 \times {2^3}$  

672${x^6}$   A1     N3

[4 marks]

valid approach     (M1)

eg$\,\,\,\,\,$recognizing ${x^7}$ is found when multiplying $5x \times 672{x^6}$

$3360{x^7}$     A1     N2

[2 marks]

Many candidates approached this question using an appropriate and efficient method to identify the required term. While many of those who were successful in (a) were also successful in (b), a significant number of candidates did not realize that multiplying their answer in (a) by $5x$ would give them the term in ${x^7}$. This led to an attempt to find a binomial expansion, which was generally unsuccessful. Some candidates continue to be unable to distinguish between the “coefficient” and the “term” and lost a point as a result.

Many candidates approached this question using an appropriate and efficient method to identify the required term. While many of those who were successful in (a) were also successful in (b), a significant number of candidates did not realize that multiplying their answer in (a) by $5x$ would give them the term in ${x^7}$. This led to an attempt to find a binomial expansion, which was generally unsuccessful. Some candidates continue to be unable to distinguish between the “coefficient” and the “term” and lost a point as a result.