| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Prove sum formula |
| Difficulty | Moderate -0.8 This is a standard C2 question testing routine geometric series knowledge. Part (a) is a bookwork proof of the standard formula, part (b) involves solving simultaneous equations with given terms, and part (c) applies the sum to infinity formula. All techniques are straightforward applications of memorized formulas with no novel problem-solving required. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \(S = a + ar + ar^2 + \ldots + ar^{n-1}\) | B1 | |
| \(rS = ar + ar^2 + \ldots + ar^n\) | M1 | |
| Subtract: \(S(1-r) = a(1-r^n)\), \(S = \dfrac{a(1-r^n)}{1-r}\) | M1 A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \(ar = 3\), \(ar^3 = 1.08\) | B1 B1 | |
| Divide: \(r^2 = 0.36\), \(r = 0.6\) | M1 A1 | |
| \(a = 6 \div 1.2 = 5\) | A1 | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Notes |
| \(S = \dfrac{5}{1-0.6}\) | M1 A1ft | |
| \(= 12.5\) | A1 | (3 marks) |
## Question 7:
### Part (a):
| Working | Mark | Notes |
|---------|------|-------|
| $S = a + ar + ar^2 + \ldots + ar^{n-1}$ | B1 | |
| $rS = ar + ar^2 + \ldots + ar^n$ | M1 | |
| Subtract: $S(1-r) = a(1-r^n)$, $S = \dfrac{a(1-r^n)}{1-r}$ | M1 A1 | (4 marks) |
### Part (b):
| Working | Mark | Notes |
|---------|------|-------|
| $ar = 3$, $ar^3 = 1.08$ | B1 B1 | |
| Divide: $r^2 = 0.36$, $r = 0.6$ | M1 A1 | |
| $a = 6 \div 1.2 = 5$ | A1 | (5 marks) |
### Part (c):
| Working | Mark | Notes |
|---------|------|-------|
| $S = \dfrac{5}{1-0.6}$ | M1 A1ft | |
| $= 12.5$ | A1 | (3 marks) |
---7. A geometric series is $a + a r + a r ^ { 2 } + \ldots$
\begin{enumerate}[label=(\alph*)]
\item Prove that the sum of the first $n$ terms of this series is given by $S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$.
The second and fourth terms of the series are 3 and 1.08 respectively.\\
Given that all terms in the series are positive, find
\item the value of $r$ and the value of $a$,
\item the sum to infinity of the series.\\
\includegraphics[max width=\textwidth, alt={}, center]{1033051d-18bf-4734-a556-4c8e1c789992-4_764_1159_294_299}
Fig. 2 shows part of the curve with equation $y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$. The curve touches the $x$-axis at $A$ and has a maximum turning point at $B$.\\
(a) Show that the equation of the curve may be written as $y = x ( x - 3 ) ^ { 2 }$, and hence write down the coordinates of $A$.\\
(b) Find the coordinates of $B$.
The shaded region $R$ is bounded by the curve and the $x$-axis.\\
(c) Find the area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [12]}}