| Date | May 2013 | Marks available | 2 | Reference code | 13M.2.hl.TZ2.5 |
| Level | HL only | Paper | 2 | Time zone | TZ2 |
| Command term | Find | Question number | 5 | Adapted from | N/A |
Question
The arithmetic sequence \(\{ {u_n}:n \in {\mathbb{Z}^ + }\} \) has first term \({u_1} = 1.6\) and common difference d = 1.5. The geometric sequence \(\{ {v_n}:n \in {\mathbb{Z}^ + }\} \) has first term \({v_1} = 3\) and common ratio r = 1.2.
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.
Markscheme
\({u_n} - {v_n} = 1.6 + (n - 1) \times 1.5 - 3 \times {1.2^{n - 1}}{\text{ }}( = 1.5n + 0.1 - 3 \times {1.2^{n - 1}})\) 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]
Examiners report
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 \({u_n} - {v_n}\).
