Find the time, in seconds, Paco takes to run his fifth lap.

Paco runs his last lap in 260 seconds.

Find how many laps he has run on Monday.

Find the total time, in minutes, run by Paco on Monday.

On Wednesday Paco takes Lola to train. They both run the first lap of the track in 120 seconds. Each lap Lola runs takes 1.06 times as long as her previous lap.

Find the time, in seconds, Lola takes to run her third lap.

Find the total time, in seconds, Lola takes to run her first four laps.

Each lap Paco runs again takes him 10 seconds longer than his previous lap. After a certain number of laps Paco takes less time per lap than Lola.

Find the number of the lap when this happens.

\(120 + 10 \times 4\)     (M1)(A1)

Notes: Award (M1) for substituted AP formula, (A1) for correct substitutions. Accept a list of 4 correct terms.

= 160     (A1)(G3)

\(120 + (n - 1) \times 10 = 260\)     (M1)(M1)

Notes: Award (M1) for correctly substituted AP formula, (M1) for equating to 260. Accept a list of correct terms showing at least the 14^th and 15^th terms.

= 15     (A1)(G2)

\(\frac{{15}}{2}(120 + 260)\) or \(\frac{{15}}{2}(2 \times 120 + (15 - 1) \times 10)\)     (M1)(A1)(ft)

Notes: Award (M1) for substituted AP sum formula, (A1)(ft) for correct substitutions. Accept a sum of a list of 15 correct terms. Follow through from their answer to part (b).

2850 seconds     (A1)(ft)(G2)

Note: Award (G2) for 2850 seen with no working shown.

47.5 minutes     (A1)(ft)(G3)

Notes: A final (A1)(ft) can be awarded for correct conversion from seconds into minutes of their incorrect answer. Follow through from their answer to part (b).

\(120 \times {1.06^{3 - 1}}\)     (M1)(A1)

Notes: Award (M1) for substituted GP formula, (A1) for correct substitutions. Accept a list of 3 correct terms.

= 135 (134.832)     (A1)(G2)

\({S_4} = \frac{{120({{1.06}^4} - 1)}}{{(1.06 - 1)}}\)     (M1)(A1)

Notes: Award (M1) for substituted GP sum formula, (A1) for correct substitutions. Accept a sum of a list of 4 correct terms.

= 525 (524.953...)     (A1)(G2)

\(120 + (n - 1) \times 10 < 120 \times {1.06^{n - 1}}\)     (M1)(M1)

Notes: Award (M1) for correct left hand side, (M1) for correct right hand side. Accept an equation. Follow through from their expressions given in parts (a) and (d).

OR

List of at least 2 terms for both sequences (120, 130, … and 120, 127.2, …)     (M1)

List of correct 12^th and 13^th terms for both sequences (..., 230, 240 and …, 227.8, 241.5)     (M1)

OR

A sketch with a line and an exponential curve,     (M1)

An indication of the correct intersection point     (M1)

13^th lap     (A1)(ft)(G2)

Note: Do not award the final (A1)(ft) if final answer is not a positive integer.