| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 14 |
| Topic | Arithmetic Sequences and Series |
| Type | Mixed arithmetic and geometric |
| Difficulty | Standard +0.3 This is a multi-part question covering standard A-level sequences and series topics. Part (i) requires setting up and solving an inequality with arithmetic series formulas (routine). Part (ii) involves standard geometric progression calculations including finding terms, using logarithms for an index, and sum to infinity. Part (iii) tests understanding of sequence terminology through straightforward analysis. While comprehensive, all components use direct application of formulas and standard techniques without requiring novel insight or complex problem-solving. |
| Spec | 1.04f Sequence types: increasing, decreasing, periodic1.04h 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 |
(i) Use of the formula $S_n = \frac{1}{2}n(2a + (n-1)d)$ for at least one arithmetic series [M1]
Obtain $\frac{1}{2}n(2 + (n-1)\times 2) > \frac{1}{2}n(200 + (n-1)\times(-3))$ [A1]
Attempt to process resulting linear inequality in $n$ [M1]
Obtain $n = 41$ [A1] **4 marks**
(ii)(a) Find the common ratio $r = \frac{2-\sqrt{2}}{3\sqrt{2}}$ [B1]
Third term is $(2-\sqrt{2}) \times r = \frac{(2-\sqrt{2})\times(2-\sqrt{2})}{3\sqrt{2}} = \frac{6-4\sqrt{2}}{3\sqrt{2}} = \sqrt{2} - \frac{4}{3}$ [B1 (AG)] **2 marks**
(b) Attempt to solve $ar^{n-1} < 0.01$ [M1]
Use of logs or trial and improvement [M1]
Obtain $n = 5$ [A1] **3 marks**
(c) Use the sum to infinity formula correctly [M1]
Obtain $\frac{3\sqrt{2}}{1 - \frac{2-\sqrt{2}}{3\sqrt{2}}} = \frac{9}{2\sqrt{2}-1}$ [A1] **2 marks**
(iii)(a) Convergent [B1]
(b) Alternating, convergent [B1]
(c) Alternating, periodic [B1] **3 marks**8 (i) The sum of the first $n$ terms of the arithmetic series $1 + 3 + 5 + \ldots$ exceeds the sum of the first $n$ terms of the arithmetic series $100 + 97 + 94 + \ldots$. Find the least possible value of $n$.\\
(ii) $3 \sqrt { 2 }$ and $2 - \sqrt { 2 }$ are the first two terms of a geometric progression.
\begin{enumerate}[label=(\alph*)]
\item Show that the third term is $\sqrt { 2 } - \frac { 4 } { 3 }$.
\item Find the index $n$ of the first term that is less than 0.01.
\item Find the exact value of the sum to infinity of this progression.\\
(iii) Which of the terms 'alternating', 'periodic', 'convergent' apply to the sequences generated by the following $n$th terms, where $n$ is a positive integer?\\
(a) $1 - \left( \frac { 3 } { 4 } \right) ^ { n }$\\
(b) $\frac { 1 } { n } \cos n \pi$\\
(c) $\sec n \pi$
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q8 [14]}}