| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find sum to infinity |
| Difficulty | Moderate -0.3 This is a straightforward multi-part geometric series question requiring standard formulas and calculator work. Parts (a)-(c) involve routine algebraic manipulation (finding r from ar³/ar = ratio, then finding a, then applying sum formula), while part (d) tests understanding that S∞ - S₅₀ exists when |r| < 1. All techniques are standard C2 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = 0.9\) | M1, A1 | (2 marks) |
| (b) \(a = \frac{7.2}{(a)} = 8\) | M1, A1 | (2 marks) |
| (c) \(s_{50} = \frac{8(1-(0.9)^{50})}{1-0.9} = 79.588\) | (3dp) | M1, A1 c.a.o |
| Answer | Marks | Guidance |
|---|---|---|
| \(s_\infty - s_{50} = 80 - (c) = 0.412\) | (Awrt 3 dp) | M1, A1 |
**(a)** $ar = 7.2, ar^3 = 5.832 \Rightarrow r^2 = \frac{5.832}{7.2} (= 0.81)$
$r = 0.9$ | M1, A1 | (2 marks)
**(b)** $a = \frac{7.2}{(a)} = 8$ | M1, A1 | (2 marks)
**(c)** $s_{50} = \frac{8(1-(0.9)^{50})}{1-0.9} = 79.588$ | (3dp) | M1, A1 c.a.o | (2 marks)
**(d)** $s_\infty = \frac{8}{1-0.9} (= 80)$
$s_\infty - s_{50} = 80 - (c) = 0.412$ | (Awrt 3 dp) | M1, A1 | (2 marks)
**(Subtotal: 8 marks)**
Guidance: (a) M1: for full method $\to r^2$ or $r$. N.B. $ar^2 = 7.2, ar^4 = 5.832 \to r = 0.9$ scores M1A1 in part (a) but probably M0A0 in (b).
(c) M1: their "$a$", "$r$" in $s_{50}$ formula. (d) M1: their "$a$", "$r$" in $s_\infty$. A1: for $80 -$ their (c), i.e., their (c) only.\begin{enumerate}
\item The second and fourth terms of a geometric series are 7.2 and 5.832 respectively.
\end{enumerate}
The common ratio of the series is positive.\\
For this series, find\\
(a) the common ratio,\\
(b) the first term,\\
(c) the sum of the first 50 terms, giving your answer to 3 decimal places,\\
(d) the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.\\
\hfill \mbox{\textit{Edexcel C2 2005 Q6 [8]}}