Find an expression for ${u_n} - {v_n}$ in terms of n.

Determine the set of values of n for which ${u_n} > {v_n}$.

Determine the greatest value of ${u_n} - {v_n}$. Give your answer correct to four significant figures.

${u_n} - {v_n} = 1.6 + (n - 1) \times 1.5 - 3 \times {1.2^{n - 1}}{\text{ }}( = 1.5n + 0.1 - 3 \times {1.2^{n - 1}})$     A1A1

[2 marks]

attempting to solve ${u_n} > {v_n}$ numerically or graphically.     (M1)

$n = 2.621 \ldots ,9.695 \ldots$     (A1)

So $3 \leqslant n \leqslant 9$     A1

[3 marks]

The greatest value of ${u_n} - {v_n}$ is 1.642.     A1

Note: Do not accept 1.64.

[1 mark]

In part (a), most candidates were able to express ${u_n}$ and ${v_n}$ correctly and hence obtain a correct expression for ${u_n} - {v_n}$. Some candidates made careless algebraic errors when unnecessarily simplifying ${u_n}$ while other candidates incorrectly stated ${v_n}$ as $3{(1.2)^n}$.

In parts (b) and (c), most candidates treated n as a continuous variable rather than as a discrete variable. Candidates should be aware that a GDC’s table feature can be extremely useful when attempting such question types.

In parts (b) and (c), most candidates treated n as a continuous variable rather than as a discrete variable. Candidates should be aware that a GDC’s table feature can be extremely useful when attempting such question types. In part (c), a number of candidates attempted to find the maximum value of n rather than attempting to find the maximum value of ${u_n} - {v_n}$.