Prachi is on vacation in the United States. She is visiting the Grand Canyon.

When she reaches the top, she drops a coin down a cliff. The coin falls down a distance of \(5\) metres during the first second, \(15\) metres during the next second, \(25\) metres during the third second and continues in this way. The distances that the coin falls during each second forms an arithmetic sequence.

(i)     Write down the common difference, \(d\) , of this arithmetic sequence.

(ii)    Write down the distance the coin falls during the fourth second.

Calculate the distance the coin falls during the \(15{\text{th}}\) second.

Calculate the total distance the coin falls in the first \(15\) seconds. Give your answer in kilometres.

Prachi drops the coin from a height of \(1800\) metres above the ground.

Calculate the time, to the nearest second, the coin will take to reach the ground.

Prachi visits a tourist centre nearby. It opened at the start of \(2015\) and in the first year there were \(17\,000\) visitors. The number of people who visit the tourist centre is expected to increase by \(10\,\% \) each year.

Calculate the number of people expected to visit the tourist centre in \(2016\).

Calculate the total number of people expected to visit the tourist centre during the first \(10\) years since it opened.

(i)     \(10\,({\text{m}})\)          (A1)

(ii)    \(35\,({\text{m}})\)          (A1)(ft)

Note: Follow through from part (a)(i).

\(5 + 14 \times 10\)        (M1)

Note: Award (M1) for correct substitution into arithmetic sequence formula. A list of their \(10\) correct terms (excluding those given in question and the \(35\) from part (a)(ii)) must be seen for the (M1) to be awarded.

\( = 145\,({\text{m}})\)         (A1)(ft)(G2)

Note: Follow through from their value for \(d\).

If a list is used, award (A1) for their \({15^{{\text{th}}}}\) term.

\(\frac{{15}}{2}(2 \times 5 + 14 \times 10)\) OR \(\frac{{15}}{2}(5 + 145)\)         (M1)

Note: Award (M1) for correct substitution into arithmetic series formula. Follow through from their part (a)(i). Accept a list added together until the \(15{\text{th}}\) term.

\( = 1125\,\,{\text{(m)}}\)          (A1)(ft)

Note: Follow through from parts (a) and (b).

\({\text{ = 1}}{\text{.13}}\,\,{\text{(km)}}\,\,(1.125\,{\text{(km)}})\)          (A1)(ft)(G2)    

Note: Award (A1)(ft) for correctly converting their metres to kilometres, irrespective of method used. To award the last (A1)(ft) in follow through, the candidate’s answer in metres must be seen.

\(\frac{n}{2}\left( {2 \times 5 + (n - 1)10} \right) = 1800\)           (M1)

Note: Award (M1) for correct substitution into arithmetic series formula equated to \(1800\). Follow through from their part (a)(i). Accept a list of terms that shows clearly the \(18{\text{th}}\) second and \(19{\text{th}}\) second distances.
Correct use of kinematics equations is a valid method.

\(n = 18.97\)         (A1)(ft)

\(19\) (seconds)         (A1)(ft)(G2)

Note: Award (A1)(ft) for correct unrounded value for \(n\). The second (A1)(ft) is awarded for the correct rounding off of their value for \(n\) to the nearest second if their unrounded value is seen.
Award (M1)(A2)(ft) for their \(19\) if method is shown. Unrounded value for \(n\) may not be seen. Follow through from their \({u_I}\) and \(d\) only if workings are shown.

OR

\(1125 + 155 + 165 + 175 + 185 = 1805\)           (M1)

Note: Award (M1) for adding the terms until reaching \(1800\).

\((n = )\,19\)            (A2)(ft)

Note: In this method, follow through from their \(d\) from part (a) and their \(1125\) from part (c).

\(17\,000\,\,(1.1)\) (or equivalent)          (M1)

Note: Award (M1) for multiplying \(17\,000\) by \(1.1\) or equivalent.

\( = 18\,700\)                   (A1)(G2)

\({S_{10}} = \frac{{17\,000\,({{1.1}^{10}} - 1)}}{{1.1 - 1}}\)           (M1)(A1)(ft)

Note: Award (M1) for substitution into the geometric series formula, (A1)(ft) for correct substitution. Award (A1)(ft) for a list of their correct \(10\) terms, (M1) for adding their \(10\) terms.

\(271\,000\,\,\,(270\,936)\)           (A1)(ft)(G2)

Note: Follow through from their \(1.1\) in part (e).

Question 2: Arithmetic and geometric sequences and series
Parts (a), (b), (c) and (e) were well done. Quite a few forgot to convert their answer to km in part (c). The main problem with part (d) was that candidates chose to equate the \({n^{{\text{th}}}}\) term formula to 1800 rather than the sum of the first n terms formula. Some of those who managed to write the correct equation were not always successful at solving it. Some candidates made use of the trial and error method to reach the correct answer. Part (e) was obvious to some, others put it into a formula with little understanding and a surprising number of candidates had place value issues (stating 10% of 17000 was 170). Many candidates used the compound interest formula in both parts (e) and (f). In part (f) many candidates did not realize that they needed to use the sum of a geometric series formula. They either used the sum of an arithmetic series or as previously mentioned, the compound interest formula.

Question 2: Arithmetic and geometric sequences and series

Parts (a), (b), (c) and (e) were well done. Quite a few forgot to convert their answer to km in part (c). The main problem with part (d) was that candidates chose to equate the \({n^{{\text{th}}}}\) term formula to 1800 rather than the sum of the first n terms formula. Some of those who managed to write the correct equation were not always successful at solving it. Some candidates made use of the trial and error method to reach the correct answer. Part (e) was obvious to some, others put it into a formula with little understanding and a surprising number of candidates had place value issues (stating 10% of 17000 was 170). Many candidates used the compound interest formula in both parts (e) and (f). In part (f) many candidates did not realize that they needed to use the sum of a geometric series formula. They either used the sum of an arithmetic series or as previously mentioned, the compound interest formula.

Question 2: Arithmetic and geometric sequences and series

Parts (a), (b), (c) and (e) were well done. Quite a few forgot to convert their answer to km in part (c). The main problem with part (d) was that candidates chose to equate the \({n^{{\text{th}}}}\) term formula to 1800 rather than the sum of the first n terms formula. Some of those who managed to write the correct equation were not always successful at solving it. Some candidates made use of the trial and error method to reach the correct answer. Part (e) was obvious to some, others put it into a formula with little understanding and a surprising number of candidates had place value issues (stating 10% of 17000 was 170). Many candidates used the compound interest formula in both parts (e) and (f). In part (f) many candidates did not realize that they needed to use the sum of a geometric series formula. They either used the sum of an arithmetic series or as previously mentioned, the compound interest formula.

Question 2: Arithmetic and geometric sequences and series

Parts (a), (b), (c) and (e) were well done. Quite a few forgot to convert their answer to km in part (c). The main problem with part (d) was that candidates chose to equate the \({n^{{\text{th}}}}\) term formula to 1800 rather than the sum of the first n terms formula. Some of those who managed to write the correct equation were not always successful at solving it. Some candidates made use of the trial and error method to reach the correct answer. Part (e) was obvious to some, others put it into a formula with little understanding and a surprising number of candidates had place value issues (stating 10% of 17000 was 170). Many candidates used the compound interest formula in both parts (e) and (f). In part (f) many candidates did not realize that they needed to use the sum of a geometric series formula. They either used the sum of an arithmetic series or as previously mentioned, the compound interest formula.

Question 2: Arithmetic and geometric sequences and series

Parts (a), (b), (c) and (e) were well done. Quite a few forgot to convert their answer to km in part (c). The main problem with part (d) was that candidates chose to equate the \({n^{{\text{th}}}}\) term formula to 1800 rather than the sum of the first n terms formula. Some of those who managed to write the correct equation were not always successful at solving it. Some candidates made use of the trial and error method to reach the correct answer. Part (e) was obvious to some, others put it into a formula with little understanding and a surprising number of candidates had place value issues (stating 10% of 17000 was 170). Many candidates used the compound interest formula in both parts (e) and (f). In part (f) many candidates did not realize that they needed to use the sum of a geometric series formula. They either used the sum of an arithmetic series or as previously mentioned, the compound interest formula.

Question 2: Arithmetic and geometric sequences and series

Parts (a), (b), (c) and (e) were well done. Quite a few forgot to convert their answer to km in part (c). The main problem with part (d) was that candidates chose to equate the \({n^{{\text{th}}}}\) term formula to 1800 rather than the sum of the first n terms formula. Some of those who managed to write the correct equation were not always successful at solving it. Some candidates made use of the trial and error method to reach the correct answer. Part (e) was obvious to some, others put it into a formula with little understanding and a surprising number of candidates had place value issues (stating 10% of 17000 was 170). Many candidates used the compound interest formula in both parts (e) and (f). In part (f) many candidates did not realize that they needed to use the sum of a geometric series formula. They either used the sum of an arithmetic series or as previously mentioned, the compound interest formula.