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

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

Find the value of $k$.

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

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

Calculate the total distance Carlos runs in the first year.

3800 m     (A1)

[1 mark]

$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]

$3000 + (k - 1)400 > 42195$     (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.

OR

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

OR

$7.5 \times {1.2^{5 - 1}}$     (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).

$= 297\,000{\text{ m }}(296\,853 \ldots {\text{ m}},{\text{ }}297{\text{ km}})$     (A1)(G2)

[3 marks]