| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 11 |
| 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 basic geometric series knowledge. Part (a) is a routine proof from first principles that all students learn, parts (b-c) are direct formula applications, and part (d) is simple recall. The proof requires no insight beyond the standard algebraic manipulation (multiply by r and subtract), making this easier than average. |
| 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(S_n = a + ar + \ldots + ar^{n-1}\) | B1 | At least one \(+\) sign, \(a\), \(ar^{n-1}\) and one other intermediate term. No extra terms (usually \(ar^n\)) |
| \(rS_n = ar + ar^2 + \ldots + ar^n\) | M1 | Multiply by \(r\); at least 2 of their terms on RHS correctly multiplied by \(r\) |
| \((1-r)S_n = a(1-r^n)\) | dM1 | Subtract both sides: LHS must be \(\pm(1-r)S_n\), RHS in form \(\pm a(1-r^{pn+q})\). Only award if \(S_n = \ldots\) or \(rS_n = \ldots\) contains term of form \(ar^{cn+d}\) |
| \(S_n = \dfrac{a(1-r^n)}{1-r}\) | A1 cso | Answer given in question; completion c.s.o. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a = 200,\ r = 2,\ n = 10\), substituted into \(S_n\) formula | M1 | Substitute \(r=2\) with \(a=100\) or \(200\) and \(n=9\) or \(10\) |
| \(S_{10} = \dfrac{200(1-2^{10})}{1-2}\) | A1 | Or equivalent |
| \(= 204{,}600\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a = \dfrac{5}{6},\ r = \dfrac{1}{3}\) | B1 | \(r = \dfrac{1}{3}\) seen or implied anywhere |
| \(S_\infty = \dfrac{a}{1-r} = \dfrac{\frac{5}{6}}{1-\frac{1}{3}}\) | M1 | Substitute \(a=\dfrac{5}{6}\) and their \(r\) into \(\dfrac{a}{1-r}\); usual rules about quoting formula |
| \(= \dfrac{5}{4}\) | A1 | o.e. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(-1 < r < 1\) (or \(\ | r\ | < 1\)) |
# Question 10:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $S_n = a + ar + \ldots + ar^{n-1}$ | B1 | At least one $+$ sign, $a$, $ar^{n-1}$ and one other intermediate term. No extra terms (usually $ar^n$) |
| $rS_n = ar + ar^2 + \ldots + ar^n$ | M1 | Multiply by $r$; at least 2 of their terms on RHS correctly multiplied by $r$ |
| $(1-r)S_n = a(1-r^n)$ | dM1 | Subtract both sides: LHS must be $\pm(1-r)S_n$, RHS in form $\pm a(1-r^{pn+q})$. Only award if $S_n = \ldots$ or $rS_n = \ldots$ contains term of form $ar^{cn+d}$ |
| $S_n = \dfrac{a(1-r^n)}{1-r}$ | A1 cso | Answer given in question; completion c.s.o. |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a = 200,\ r = 2,\ n = 10$, substituted into $S_n$ formula | M1 | Substitute $r=2$ with $a=100$ or $200$ and $n=9$ or $10$ |
| $S_{10} = \dfrac{200(1-2^{10})}{1-2}$ | A1 | Or equivalent |
| $= 204{,}600$ | A1 | |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a = \dfrac{5}{6},\ r = \dfrac{1}{3}$ | B1 | $r = \dfrac{1}{3}$ seen or implied anywhere |
| $S_\infty = \dfrac{a}{1-r} = \dfrac{\frac{5}{6}}{1-\frac{1}{3}}$ | M1 | Substitute $a=\dfrac{5}{6}$ and their $r$ into $\dfrac{a}{1-r}$; usual rules about quoting formula |
| $= \dfrac{5}{4}$ | A1 | o.e. |
## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $-1 < r < 1$ (or $\|r\| < 1$) | B1 (1) | In words or symbols. Must use $<$ not $\leq$. If words and symbols contradictory, take symbols |\begin{enumerate}
\item A geometric series is $a + a r + a r ^ { 2 } + \ldots$\\
(a) Prove that the sum of the first $n$ terms of this series is given by
\end{enumerate}
$$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$
(b) Find
$$\sum _ { k = 1 } ^ { 10 } 100 \left( 2 ^ { k } \right)$$
(c) Find the sum to infinity of the geometric series
$$\frac { 5 } { 6 } + \frac { 5 } { 18 } + \frac { 5 } { 54 } + \ldots$$
(d) State the condition for an infinite geometric series with common ratio $r$ to be convergent.
\hfill \mbox{\textit{Edexcel C2 2007 Q10 [11]}}