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Date	November 2017	Marks available	2	Reference code	17N.2.SL.TZ0.T_2
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	T_2	Adapted from	N/A

Question

Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres.

A marathon is 42.195 kilometres.

In the $k$ th training session Rosa will run further than a marathon for the first time.

Carlos joins the club to lose weight. He runs 7500 metres during the first month. The distance he runs increases by 20% each month.

Write down the distance Rosa runs in the third training session;

[1]

a.i.

Write down the distance Rosa runs in the $n$ th training session.

[2]

a.ii.

Find the value of $k$ .

[2]

Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.

[4]

Find the distance Carlos runs in the fifth month of training.

[3]

Calculate the total distance Carlos runs in the first year.

[3]

Markscheme

3800 m (A1)

[1 mark]

a.i.

$3000 + (n - 1)400{\text{ m}}$ $\,\,\,$ OR $\,\,\,$ $2600 + 400n{\text{ m}}$ (M1)(A1)

Note: Award (M1) for substitution into arithmetic sequence formula, (A1) for correct substitution.

[2 marks]

a.ii.

$3000 + (k - 1)400 > 42195$ (M1)

Notes: Award (M1) for their correct inequality. Accept $3 + (k - 1)0.4 > 42.195$ .

Accept $=$ OR $\geqslant$ . Award (M0) for $3000 + (k - 1)400 > 42.195$ .

$(k = ){\text{ }}99$ (A1)(ft)(G2)

Note: Follow through from part (a)(ii), but only if $k$ is a positive integer.

[2 marks]

$\frac{{50}}{2}\left( {2 \times 3000 + (50 - 1)(400)} \right)$ (M1)(A1)(ft)

Note: Award (M1) for substitution into sum of an arithmetic series formula, (A1)(ft) for correct substitution.

$640\,000{\text{ m}}$ (A1)

Note: Award (A1) for their $640\,000$ seen.

$= 640{\text{ km}}$ (A1)(ft)(G3)

Note: Award (A1)(ft) for correctly converting their answer in metres to km; this can be awarded independently from previous marks.

$\frac{{50}}{2}\left( {2 \times 3 + (50 - 1)(0.4)} \right)$ (M1)(A1)(ft)(A1)

Note: Award (M1) for substitution into sum of an arithmetic series formula, (A1)(ft) for correct substitution, (A1) for correctly converting 3000 m and 400 m into km.

$= 640{\text{ km}}$ (A1)(G3)

[4 marks]

$7500 \times {1.2^{5 - 1}}$ (M1)(A1)

Note: Award (M1) for substitution into geometric series formula, (A1) for correct substitutions.

$= 15\,600{\text{ m }}(15\,552{\text{ m}})$ (A1)(G3)

$7.5 \times {1.2^{5 - 1}}$ (M1)(A1)

Note: Award (M1) for substitution into geometric series formula, (A1) for correct substitutions.

$= 15.6{\text{ km}}$ (A1)(G3)

[3 marks]

$\frac{{7500({{1.2}^{12}} - 1)}}{{1.2 - 1}}$ (M1)(A1)

Notes: Award (M1) for substitution into sum of a geometric series formula, (A1) for correct substitutions. Follow through from their ratio ( $r$ ) in part (d). If $r < 1$ (distance does not increase) or the final answer is unrealistic (eg $r = 20$ ), do not award the final (A1).