Ramiro walks to work each morning. During the first minute he walks $80$ metres. In each subsequent minute he walks $90\%$ of the distance walked during the previous minute.

The distance between his house and work is $660$ metres. Ramiro leaves his house at 08:00 and has to be at work by 08:15.

Explain why he will not be at work on time.

METHOD 1

recognize that the distance walked each minute is a geometric sequence     (M1)

eg$\;\;\;r = 0.9$, valid use of $0.9$

recognize that total distance walked is the sum of a geometric sequence     (M1)

eg$\;\;\;{S_n},{\text{ }}a\left( {\frac{{1 - {r^n}}}{{1 - r}}} \right)$

correct substitution into the sum of a geometric sequence     (A1)

eg$\;\;\;80\left( {\frac{{1 - {{0.9}^n}}}{{1 - 0.9}}} \right)$

any correct equation with sum of a geometric sequence     (A1)

eg$\;\;\;80\left( {\frac{{{{0.9}^n} - 1}}{{0.9 - 1}}} \right) = 660,{\text{ }}1 - {0.9^n} = \frac{{66}}{{80}}$

attempt to solve their equation involving the sum of a GP     (M1)

eg$\;\;\;$graph, algebraic approach

$n = 16.54290788$     A1

since $n > 15$     R1

he will be late     AG     N0

Note:     Do not award the R mark without the preceding A mark.

METHOD 2

recognize that the distance walked each minute is a geometric sequence     (M1)

eg$\;\;\;r = 0.9$, valid use of $0.9$

recognize that total distance walked is the sum of a geometric sequence     (M1)

eg$\;\;\;{S_n},{\text{ }}a\left( {\frac{{1 - {r^n}}}{{1 - r}}} \right)$

correct substitution into the sum of a geometric sequence     (A1)

eg$\;\;\;80\left( {\frac{{1 - {{0.9}^n}}}{{1 - 0.9}}} \right)$

attempt to substitute $n = 15$ into sum of a geometric sequence     (M1)

eg$\;\;\;{S_{15}}$

correct substitution     (A1)

eg$\;\;\;80\left( {\frac{{{{0.9}^{15}} - 1}}{{0.9 - 1}}} \right)$

${S_{15}} = 635.287$     A1

since $S < 660$     R1

he will not be there on time     AG     N0

Note:     Do not award the R mark without the preceding A mark.

METHOD 3

recognize that the distance walked each minute is a geometric sequence     (M1)

eg$\;\;\;r = 0.9$, valid use of $0.9$

recognize that total distance walked is the sum of a geometric sequence     (M1)

eg$\;\;\;{S_n},{\text{ }}a\left( {\frac{{1 - {r^n}}}{{1 - r}}} \right)$

listing at least 5 correct terms of the GP     (M1)

15 correct terms     A1

$80,{\rm{ }}72,{\rm{ }}64.8,{\rm{ }}58.32,{\rm{ }}52.488,{\rm{ }}47.2392,{\rm{ }}42.5152,{\rm{ }}38.2637,{\rm{ }}34.4373,{\rm{ }}30.9936,{\rm{ }}27.8942,{\rm{ }}25.1048,{\rm{ }}22.59436,{\rm{ }}20.3349,{\rm{ }}18.3014$

attempt to find the sum of the terms     (M1)

eg$\;\;\;{S_{15}},{\text{ }}80 + 72 + 64.8 + 58.32 + 52.488 +  \ldots  + 18.301433$

${S_{15}} = 635.287$     A1

since $S < 660$     R1

he will not be there on time     AG     N0

Note:     Do not award the R mark without the preceding A mark.

[7 marks]

Many found this question accessible, although the most common approach was to calculate each term by brute force, which at times contained small errors or inaccuracies that affected the overall sum. Although this was a valid method, it meant an inefficient use of time that could have affected the performance on other questions.

Those who applied the formula for geometric series were typically more successful and far more efficient in answering the question.