| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2004 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Sum of first n terms |
| Difficulty | Easy -1.3 This question involves straightforward application of standard formulas for geometric and arithmetic series with no problem-solving required. Part (i) requires identifying r=2/3 and applying the GP sum formula, while part (ii) requires finding n from the last term then applying the AP sum formula. Both are routine textbook exercises testing basic recall and calculation skills. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = 54/81\) or \(36/54\) | B1 | Value of \(r\) – unsimplified – allow 0.66 |
| \(S_{10} = 81(1 - \frac{2}{3}^{10}) \div (1 - \frac{2}{3})\) | M1 | Correct formula – power 10 and used. Co. More than 3 s.f. ok, but needs 238.8 |
| \(\rightarrow 239\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(n = (180 - 25) \div 5 + 1 = 32\) | B1 | 31 gets M0 |
| Use of any \(S_n\) formula | M1 | Correct formula – not for \(n = 25, 5, 180\) |
| \(\rightarrow 3280\) | A1 | [3] Co |
## Question 2(i): $81, 54, 36$
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = 54/81$ or $36/54$ | B1 | Value of $r$ – unsimplified – allow 0.66 |
| $S_{10} = 81(1 - \frac{2}{3}^{10}) \div (1 - \frac{2}{3})$ | M1 | Correct formula – power 10 and used. Co. More than 3 s.f. ok, but needs 238.8 |
| $\rightarrow 239$ | A1 | [3] |
## Question 2(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $n = (180 - 25) \div 5 + 1 = 32$ | B1 | 31 gets M0 |
| Use of any $S_n$ formula | M1 | Correct formula – not for $n = 25, 5, 180$ |
| $\rightarrow 3280$ | A1 | [3] Co |
---2 Find\\
(i) the sum of the first ten terms of the geometric progression $81,54,36 , \ldots$,\\
(ii) the sum of all the terms in the arithmetic progression $180,175,170 , \ldots , 25$.
\hfill \mbox{\textit{CAIE P1 2004 Q2 [6]}}