| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Topic | Arithmetic Sequences and Series |
| Type | Two related arithmetic progressions |
| Difficulty | Standard +0.3 This is a straightforward arithmetic progression question requiring basic formula application (a + 12d = 60, a + 30d = 141) to find first term and common difference, then identifying common terms. The 'proof' in part (ii)(b) simply requires showing both sequences can be expressed in forms that yield integer solutions, which is routine algebraic manipulation rather than genuine mathematical insight. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Stating both of \(60 = a + 12d\) and \(141 = a + 30d\) | M1 | |
| Obtain \(a = 6\) and \(d = 4.5\) | A1 A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| (a) The two sequences start: 6, 10.5, 15, 19.5 ... and 1.5, 4.5, 7.5, 10.5 ... | B1 | [1] |
| (b) Noting that \(6 + (2n-1)4.5 = 1.5 + (3n + 1 - 1)3\), the \(2n^{\text{th}}\) member of the first progression is identical to the \((3n+1)^{\text{th}}\) member of the second sequence. (Or equivalent convincing statement) | M1 | A1 |
### (i)
Stating both of $60 = a + 12d$ and $141 = a + 30d$ | M1 |
Obtain $a = 6$ and $d = 4.5$ | A1 A1 | [3]
### (ii)
**(a)** The two sequences start: 6, 10.5, 15, 19.5 ... and 1.5, 4.5, 7.5, 10.5 ... | B1 | [1]
**(b)** Noting that $6 + (2n-1)4.5 = 1.5 + (3n + 1 - 1)3$, the $2n^{\text{th}}$ member of the first progression is identical to the $(3n+1)^{\text{th}}$ member of the second sequence. (Or equivalent convincing statement) | M1 | A1 | [2]An arithmetic progression has 13th term equal to 60 and 31st term equal to 141.
\begin{enumerate}[label=(\roman*)]
\item Find the first term and common difference of the progression. [3]
\end{enumerate}
A second arithmetic progression has first term 1.5 and common difference 3.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item \begin{enumerate}[label=(\alph*)]
\item Write down the first four terms of each progression. [1]
\item Prove that the two progressions have an infinite number of terms in common. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2010 Q3 [6]}}