| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Mixed arithmetic and geometric |
| Difficulty | Moderate -0.8 This question tests standard formulas for geometric series (sum to infinity) and arithmetic series (sum formula), requiring straightforward substitution with r=32/36=8/9 and solving a simple quadratic equation. Both parts are routine applications of memorized formulas with no conceptual challenges or multi-step reasoning beyond basic algebra. |
| 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 |
|---|---|---|
| 36, 32, ... | ||
| (i) \(r = \frac{8}{9}\), \(S_\infty = (\text{their } a) \times (1 - \text{their } r)\) | M1 | Method for \(r\) and \(S_\infty\) ok. (\(\ |
| \(S_\infty = 36 + \frac{1}{9} = 324\) | A1 | co |
| [2] | ||
| (ii) \(d = -4\) | B1 | co |
| \(0 = \frac{n}{2}(72 + (n-1)(-4))\) | M1 | \(S_n\) formula ok and a value for \(d\) \(\left\ |
| \(\rightarrow n = 19\) | A1 | Condone \(n = 0\) but no other soln |
| [3] |
36, 32, ... | | |
(i) $r = \frac{8}{9}$, $S_\infty = (\text{their } a) \times (1 - \text{their } r)$ | M1 | Method for $r$ and $S_\infty$ ok. ($\|r\| < 1$) |
$S_\infty = 36 + \frac{1}{9} = 324$ | A1 | co |
| | [2] |
(ii) $d = -4$ | B1 | co |
$0 = \frac{n}{2}(72 + (n-1)(-4))$ | M1 | $S_n$ formula ok and a value for $d$ $\left\|\frac{8}{9}\right\|$ |
$\rightarrow n = 19$ | A1 | Condone $n = 0$ but no other soln |
| | [3] |2 The first term in a progression is 36 and the second term is 32 .\\
(i) Given that the progression is geometric, find the sum to infinity.\\
(ii) Given instead that the progression is arithmetic, find the number of terms in the progression if the sum of all the terms is 0 .
\hfill \mbox{\textit{CAIE P1 2014 Q2 [5]}}