| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find sum to infinity |
| Difficulty | Moderate -0.8 This is a straightforward application of standard geometric series formulas with no problem-solving required. Part (a) uses ar², part (b) uses S∞ = a/(1-r) with given values, and part (c) requires recognizing that summing from n=4 onwards equals S∞ minus the first three terms. All values are simple decimals requiring only basic arithmetic, making this easier than average but not trivial since part (c) requires some understanding of series manipulation. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Third term \(= 48 \times (0.6)^2\) | M1 | Correct use of GP term formula |
| \(= 17.28\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(S_\infty = \frac{a}{1-r} = \frac{48}{1-0.6}\) | M1 | Correct formula with \( |
| \(= 120\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sum_{n=4}^{\infty} u_n = S_\infty - (u_1 + u_2 + u_3)\) | M1 | Correct method: subtract first 3 terms from \(S_\infty\) |
| \(u_1 + u_2 + u_3 = 48 + 28.8 + 17.28 = 94.08\) | A1 | Correct sum of first 3 terms |
| \(= 120 - 94.08 = 25.92\) | A1 | Correct final answer |
# Question 3:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Third term $= 48 \times (0.6)^2$ | M1 | Correct use of GP term formula |
| $= 17.28$ | A1 | Correct answer |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S_\infty = \frac{a}{1-r} = \frac{48}{1-0.6}$ | M1 | Correct formula with $|r| < 1$ |
| $= 120$ | A1 | Correct answer |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sum_{n=4}^{\infty} u_n = S_\infty - (u_1 + u_2 + u_3)$ | M1 | Correct method: subtract first 3 terms from $S_\infty$ |
| $u_1 + u_2 + u_3 = 48 + 28.8 + 17.28 = 94.08$ | A1 | Correct sum of first 3 terms |
| $= 120 - 94.08 = 25.92$ | A1 | Correct final answer |3 The first term of an infinite geometric series is 48 . The common ratio of the series is 0.6 .
\begin{enumerate}[label=(\alph*)]
\item Find the third term of the series.
\item Find the sum to infinity of the series.
\item The $n$th term of the series is $u _ { n }$. Find the value of $\sum _ { n = 4 } ^ { \infty } u _ { n }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2015 Q3 [7]}}