| Exam Board | AQA |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Convergence conditions |
| Difficulty | Standard +0.3 This is a straightforward geometric series question with standard applications. Part (a) requires simple manipulation of area ratios to find side lengths (√(1/2) scaling). Part (b) involves recognizing a geometric series with r=1/√2, summing to infinity, and showing the limit is less than 3.5w—routine A-level series work. Part (c) asks for conceptual explanation of adding constant gaps, which is accessible reasoning. The question is slightly easier than average due to clear structure, standard techniques, and generous mark allocation for the difficulty level. |
| Spec | 1.02z Models in context: use functions in modelling1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks |
|---|---|
| 9(a) | w 2w |
| Answer | Marks | Guidance |
|---|---|---|
| ACF | AO1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(b) | Models the lengths as a geometric | |
| sequence | AO3.3 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| their r < 1 | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| correct sum in terms of w | AO3.4 | A1 |
| Compares their sum with 3.5w | AO2.4 | E1 |
| Answer | Marks |
|---|---|
| 9(c) | Explains that the model would have |
| Answer | Marks | Guidance |
|---|---|---|
| each tile | AO3.5c | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3.5w | AO3.5a | E1 |
| Total | 7 | |
| Q | Marking Instructions | AO |
Question 9:
--- 9(a) ---
9(a) | w 2w
Obtains correct length =
2 2
ACF | AO1.1b | B1 | w
2
--- 9(b) ---
9(b) | Models the lengths as a geometric
sequence | AO3.3 | M1 | 1
a=w and r =
2
w
S =
∞ 1
1−
2
≈3.41w<3.5w
Finds the sum to infinity provided
their r < 1 | AO1.1a | M1
Uses their model to obtain the
correct sum in terms of w | AO3.4 | A1
Compares their sum with 3.5w | AO2.4 | E1
--- 9(c) ---
9(c) | Explains that the model would have
to include an additional 3 mm for
each tile | AO3.5c | E1 | The total length will now include an
additional 3 mm for each tile.
The total length will not have an
upper limit.
Explains that the total length will
not have an upper limit
Or
The total length may now exceed
3.5w | AO3.5a | E1
Total | 7
Q | Marking Instructions | AO | Marks | Typical SolutionHelen is creating a mosaic pattern by placing square tiles next to each other along a straight line.
\includegraphics{figure_9}
The area of each tile is half the area of the previous tile, and the sides of the largest tile have length $w$ centimetres.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $w$, the length of the sides of the second largest tile.
[1 mark]
\item Assume the tiles are in contact with adjacent tiles, but do not overlap.
Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than $3.5w$.
[4 marks]
\item Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles.
Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 2018 Q9 [7]}}