| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find sum to infinity |
| Difficulty | Moderate -0.8 This is a straightforward geometric progression question requiring standard techniques: setting up equations from given terms (ar=18, ar³=8), solving for r and a, then applying the sum to infinity formula. All steps are routine with no conceptual challenges beyond basic GP knowledge. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(ar = 18\) and \(ar^3 = 8\). Solution to give \(r = 2/3\). \(a = 18 ÷ r = 27.0\) | M1, DM1, A1 | Any 2 equations of type \(ar^n\); Correct method on correct 2 equations; For his \(18 + r\) |
| (ii) Sum to infinity = \(a ÷ (1-r)\). Answer = 81.0 | M1, A1∨ | Correct formula applied – even if \(r > 1\); Follow through provided \(r < 1\) (ignore \(r = ±2/3\)) |
**(i)** $ar = 18$ and $ar^3 = 8$. Solution to give $r = 2/3$. $a = 18 ÷ r = 27.0$ | M1, DM1, A1 | Any 2 equations of type $ar^n$; Correct method on correct 2 equations; For his $18 + r$
**(ii)** Sum to infinity = $a ÷ (1-r)$. Answer = 81.0 | M1, A1∨ | Correct formula applied – even if $r > 1$; Follow through provided $r < 1$ (ignore $r = ±2/3$)2 A geometric progression, for which the common ratio is positive, has a second term of 18 and a fourth term of 8 . Find\\
(i) the first term and the common ratio of the progression,\\
(ii) the sum to infinity of the progression.
\hfill \mbox{\textit{CAIE P1 2002 Q2 [5]}}