| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Show quadratic equation in n |
| Difficulty | Moderate -0.3 This is a straightforward application of standard arithmetic progression formulas (S_n and nth term). Part (a) requires algebraic manipulation of the sum formula to reach a given result, while part (b) involves solving simultaneous equations. The question is slightly easier than average as it's purely procedural with no problem-solving insight required, though the algebra is non-trivial enough to avoid being trivial. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{n}{2}[8+(n-1)d]=5863\) leading to \(n[8+(n-1)d]=11726\) | B1 | Must show a useful intermediate step. WWW AG. |
| leading to \((n-1)d=\frac{11726}{n}-8\) | 1 total |
| Answer | Marks | Guidance |
|---|---|---|
| \(4+(n-1)d=139\) leading to \(\frac{11726}{n}-8=135\) | \*M1 | OE Use of correct \(u_n\) formula with expression from (a) or \(S_n\) formula to eliminate \(d\) |
| \(n=\frac{11726}{143}=82\) | A1 | |
| \(81d=\frac{11726}{82}-8\) | DM1 | Substitute *their* \(n\) into a correct \(u_n\) or \(S_n\) formula |
| \(d=\frac{5}{3}\) | A1 | Accept \(\frac{138}{81}\) OE fraction only. If M0 DM0 scored then SC B1 B1 for correct \(n\) and \(d\) values only. |
## Question 3(a):
$\frac{n}{2}[8+(n-1)d]=5863$ leading to $n[8+(n-1)d]=11726$ | **B1** | Must show a useful intermediate step. WWW AG.
leading to $(n-1)d=\frac{11726}{n}-8$ | **1 total** |
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## Question 3(b):
$4+(n-1)d=139$ leading to $\frac{11726}{n}-8=135$ | **\*M1** | OE Use of correct $u_n$ formula with expression from (a) or $S_n$ formula to eliminate $d$
$n=\frac{11726}{143}=82$ | **A1** |
$81d=\frac{11726}{82}-8$ | **DM1** | Substitute *their* $n$ into a correct $u_n$ or $S_n$ formula
$d=\frac{5}{3}$ | **A1** | Accept $\frac{138}{81}$ OE fraction only. If M0 DM0 scored then **SC B1 B1** for correct $n$ and $d$ values only.
**4 total**
---3 An arithmetic progression has first term 4 and common difference $d$. The sum of the first $n$ terms of the progression is 5863.
\begin{enumerate}[label=(\alph*)]
\item Show that $( n - 1 ) d = \frac { 11726 } { n } - 8$.
\item Given that the $n$th term is 139 , find the values of $n$ and $d$, giving the value of $d$ as a fraction.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q3 [5]}}