| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find N for S_∞ - S_N condition |
| Difficulty | Standard +0.3 Part (a) is a straightforward arithmetic progression problem requiring the sum formula and solving a quadratic. Part (b) involves manipulating the GP sum formulas (S_n and S_∞) and basic algebraic rearrangement to reach the given inequality, but the steps are fairly standard once the formulas are recalled. This is slightly above average difficulty due to the algebraic manipulation in part (b) and the need to work with inequalities, but it remains a routine textbook-style question without requiring novel insight. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(d = -7\) used | B1 | co |
| \((m/2)[322 + (m-1)(-7)] = 0\) | M1 | Condone omission of \((m/2)\); statement co (condone \(m=0\)) |
| \(47\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\dfrac{a(1-r^n)}{1-r} < \dfrac{0.9a}{1-r}\) | M1 | Allow for \(=, <, >, \leq, \geq\) |
| \(1 - r^n < 0.9\) | M1 | Needs inequality sign correct |
| \(r^n > 0.1\) | A1 | co |
## Question 5:
**Part (a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $d = -7$ used | B1 | co |
| $(m/2)[322 + (m-1)(-7)] = 0$ | M1 | Condone omission of $(m/2)$; statement co (condone $m=0$) |
| $47$ | A1 | |
**Part (b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\dfrac{a(1-r^n)}{1-r} < \dfrac{0.9a}{1-r}$ | M1 | Allow for $=, <, >, \leq, \geq$ |
| $1 - r^n < 0.9$ | M1 | Needs inequality sign correct |
| $r^n > 0.1$ | A1 | co |
---5
\begin{enumerate}[label=(\alph*)]
\item The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum of the first $m$ terms is zero. Find the value of $m$.
\item A geometric progression, in which all the terms are positive, has common ratio $r$. The sum of the first $n$ terms is less than $90 \%$ of the sum to infinity. Show that $r ^ { n } > 0.1$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2010 Q5 [6]}}