| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Recursive sequence definition |
| Difficulty | Easy -1.2 This is a straightforward geometric sequence question requiring only direct application of standard formulas. Parts (a)-(c) involve simple substitution into the recurrence relation, (d) uses the finite GP sum formula, and (e) uses the infinite GP sum formula with |r| < 1. All steps are routine with no problem-solving or insight required. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(u_2 = 24\), \(u_3 = 16\) and \(u_4 = \frac{32}{3}\) | M1, A1 | Attempt formula correctly at least twice; all three correct exact simplified answers, allow \(10.\dot{6}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = \frac{2}{3}\) | B1 | Accept \(\frac{2}{3}\) or equivalent such as \(\frac{24}{36}\); allow awrt 0.667 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(u_{11} = ar^{10} = 36 \times \left(\frac{2}{3}\right)^{10} = \left(\frac{4096}{6561}\right)\) | M1 | Uses \(u_{11} = ar^{10} = 36 \times (r)^{10}\) with their \(r\) |
| \(= 0.6243\) | A1 | Accept awrt 0.6243 or \(\frac{4096}{6561}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sum_{i=1}^{6} u_i = \frac{36\left(1-\left(\frac{2}{3}\right)^6\right)}{1-\frac{2}{3}}\) or \(36+24+16+\frac{32}{3}+u_5+u_6\) | M1 | Uses correct sum formula with \(a=36\) and their \(r\), or adds first six terms |
| \(= 98\frac{14}{27}\) | A1cao | Must be exact; \(\frac{2660}{27} = 98\frac{14}{27}\), allow \(98.\dot{5}1\dot{8}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sum_{i=1}^{\infty} u_i = \frac{36}{1-\frac{2}{3}} = 108\) | M1, A1 | Uses correct sum to infinity formula with \(a=36\) and \(r=\frac{2}{3}\) or their \(r\) as long as \( |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $u_2 = 24$, $u_3 = 16$ and $u_4 = \frac{32}{3}$ | M1, A1 | Attempt formula correctly at least twice; all three correct exact simplified answers, allow $10.\dot{6}$ |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = \frac{2}{3}$ | B1 | Accept $\frac{2}{3}$ or equivalent such as $\frac{24}{36}$; allow awrt 0.667 |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $u_{11} = ar^{10} = 36 \times \left(\frac{2}{3}\right)^{10} = \left(\frac{4096}{6561}\right)$ | M1 | Uses $u_{11} = ar^{10} = 36 \times (r)^{10}$ with their $r$ |
| $= 0.6243$ | A1 | Accept awrt 0.6243 or $\frac{4096}{6561}$ |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sum_{i=1}^{6} u_i = \frac{36\left(1-\left(\frac{2}{3}\right)^6\right)}{1-\frac{2}{3}}$ or $36+24+16+\frac{32}{3}+u_5+u_6$ | M1 | Uses correct sum formula with $a=36$ and their $r$, or adds first six terms |
| $= 98\frac{14}{27}$ | A1cao | Must be exact; $\frac{2660}{27} = 98\frac{14}{27}$, allow $98.\dot{5}1\dot{8}$ |
### Part (e):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sum_{i=1}^{\infty} u_i = \frac{36}{1-\frac{2}{3}} = 108$ | M1, A1 | Uses correct sum to infinity formula with $a=36$ and $r=\frac{2}{3}$ or their $r$ as long as $|r|<1$; answer must be exact |
---6. A sequence is defined by
$$\begin{aligned}
u _ { 1 } & = 36 \\
u _ { n + 1 } & = \frac { 2 } { 3 } u _ { n } , \quad n \geqslant 1
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Find the exact simplified values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$
\item Write down the common ratio of the sequence.
\item Find, giving your answer to 4 significant figures, the value of $u _ { 11 }$
\item Find the exact value of $\sum _ { i = 1 } ^ { 6 } u _ { i }$
\item Find the value of $\sum _ { i = 1 } ^ { \infty } u _ { i }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q6 [9]}}