| 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 | Standard +0.3 This is a standard C2 geometric series question covering formula derivation, finite sum calculation, and sum to infinity. Part (a) is a bookwork proof, parts (b-c) are routine applications with given values, and part (d) requires solving a quadratic but follows a standard pattern. Slightly above average difficulty due to the proof component and the algebraic manipulation in part (d), but all techniques are core syllabus material with no novel problem-solving required. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((S =) a + ar + ... + ar^{n-1}\) "S =" not required. Addition required. | B1 | |
| \((rS =) ar + ar^2 + ... + ar^n\) "rS =" not required (M: Multiply by r) | M1 | |
| \(S(1-r) = a(1-r^n)\) \(S = \frac{a(1-r^n)}{1-r}\) | M1, A1 | (M: Subtract and factorise each side) (*) |
| (b) \(r = 0.9\) | B1 | |
| \(S_{20} = \frac{10(1-0.9^{10})}{1-0.9} = 87.8\) | M1, A1 | (3 marks) |
| (c) Sum to infinity \(= \frac{a}{1-r} = \frac{10}{1-0.9} = 100\) | M1, A1ft | (ft only for \( |
| (d) \(\frac{a}{1-r} = \frac{r}{1-r} = 10\) (Put \(a = r\) in the formula from (c), and equate to 10) | M1 | |
| \(r = 10(1-r)\) \(r = ...\), \(\frac{10}{11}\) (or exact equivalent) | M1, A1 | (3 marks) |
**(a)** $(S =) a + ar + ... + ar^{n-1}$ "S =" not required. Addition required. | B1 |
$(rS =) ar + ar^2 + ... + ar^n$ "rS =" not required (M: Multiply by r) | M1 |
$S(1-r) = a(1-r^n)$ $S = \frac{a(1-r^n)}{1-r}$ | M1, A1 | (M: Subtract and factorise each side) (*) | (4 marks)
**(b)** $r = 0.9$ | B1 |
$S_{20} = \frac{10(1-0.9^{10})}{1-0.9} = 87.8$ | M1, A1 | (3 marks)
**(c)** Sum to infinity $= \frac{a}{1-r} = \frac{10}{1-0.9} = 100$ | M1, A1ft | (ft only for $|r| < 1$) | (2 marks)
**(d)** $\frac{a}{1-r} = \frac{r}{1-r} = 10$ (Put $a = r$ in the formula from (c), and equate to 10) | M1 |
$r = 10(1-r)$ $r = ...$, $\frac{10}{11}$ (or exact equivalent) | M1, A1 | (3 marks)A geometric series is $a + ar + ar^2 + \ldots$
\begin{enumerate}[label=(\alph*)]
\item Prove that the sum of the first $n$ terms of this series is $S_n = \frac{a(1 - r^n)}{1 - r}$. [4]
\end{enumerate}
The first and second terms of a geometric series $G$ are 10 and 9 respectively.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find, to 3 significant figures, the sum of the first twenty terms of $G$. [3]
\item Find the sum to infinity of $G$. [2]
\end{enumerate}
Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the exact value of the common ratio of this series. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [12]}}