| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Shared terms between AP and GP |
| Difficulty | Moderate -0.3 This is a straightforward two-part question requiring standard formula application. Part (i) uses S_∞ = a/(1-r) with given information to find 'a', while part (ii) applies the arithmetic series formula S_n = n/2(2a + (n-1)d). The connection between the progressions is explicit, requiring no insight beyond recognizing shared terms. Slightly easier than average due to direct formula substitution with minimal algebraic manipulation. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(ar = 3\) and \(\dfrac{a}{1-r} = 12\) | B1, B1 | co for each one |
| Solution of sim eqns \(\rightarrow a = 6\) | M1, A1 [4] | Needs to eliminate \(a\) or \(r\) correctly; co (M mark needs a quadratic) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 6\), \(d = -3\) | B1\(\sqrt{}\) | For \(d = 3 -\) his "6" |
| \(S_{20} = 10(12 - 57)\) | M1 | Sum formula must be correct and used |
| \(\rightarrow -450\) | A1 [3] | co |
## Question 7(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $ar = 3$ and $\dfrac{a}{1-r} = 12$ | B1, B1 | co for each one |
| Solution of sim eqns $\rightarrow a = 6$ | M1, A1 [4] | Needs to eliminate $a$ or $r$ correctly; co (M mark needs a quadratic) |
## Question 7(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 6$, $d = -3$ | B1$\sqrt{}$ | For $d = 3 -$ his "6" |
| $S_{20} = 10(12 - 57)$ | M1 | Sum formula must be correct and used |
| $\rightarrow -450$ | A1 [3] | co |
---7 The second term of a geometric progression is 3 and the sum to infinity is 12 .\\
(i) Find the first term of the progression.
An arithmetic progression has the same first and second terms as the geometric progression.\\
(ii) Find the sum of the first 20 terms of the arithmetic progression.
\hfill \mbox{\textit{CAIE P1 2007 Q7 [7]}}