| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Simple recurrence evaluation |
| Difficulty | Easy -1.3 This is a straightforward multi-part question testing basic recall and simple application of standard formulas. Part (i) uses the arithmetic sequence formula directly, part (ii) requires recognizing the common ratio and applying the sum to infinity formula, and part (iii) involves simple substitution into a recurrence relation with pattern recognition. All parts are routine textbook exercises requiring minimal problem-solving. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
(i) Any correct complete method; answer: 43 — M1, A1 **[2]**
(ii) $r = \frac{1}{3}$ — B1
$S_\infty = \frac{a}{1-r}$ — M1
$= \frac{162}{1 - \frac{1}{3}} = 243$ — A1 **[3]**
(iii) All four of $-1, 3, -1, 3$ — B1; It is periodic o.e. — B1 **[2]**
**Total: [7]**2 (i) An arithmetic sequence has first term 3 and common difference 2. Find the twenty-first term of this sequence.\\
(ii) Find the sum to infinity of a geometric progression with first term 162 and second term 54.\\
(iii) A sequence is given by the recurrence relation $u _ { 1 } = 3 , u _ { n + 1 } = 2 - u _ { n } , n = 1,2,3 , \ldots$. Find $u _ { 2 } , u _ { 3 }$, $u _ { 4 } , u _ { 5 }$ and describe the behaviour of this sequence.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2013 Q2 [7]}}