| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find first term from conditions |
| Difficulty | Moderate -0.3 Part (a) requires routine differentiation of the sum formula to find terms (a=31, d=-2), a standard textbook exercise. Part (b) involves setting up two equations from GP formulas (S_∞=a/(1-r)=20 and a+ar=12.8) and solving simultaneously, which requires algebraic manipulation but follows a well-practiced method. Both parts test standard techniques with no novel insight required, making this slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks |
|---|---|
| B1, M1, A1 [3] | co; Correct method. co; [M1 only when coeffs compared] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow (r = 0.6) \rightarrow a = 8\) | B1, B1, M1, DM1, A1 [5] | co co; 'Correct' elimination to form equation in \(a\) or \(r\); Complete method leading to \(a = \) Condone \(a = 8\) and 32 |
**(a)** $S_n = 32n - n^2$
Set $n$ to 1, $a$, or $S_1 = 31$
Set $n$ to 2 or other value $S_2 = 60$
$\rightarrow$ 2nd term $= 29 \rightarrow d = -2$
(or equates formulae – compares coeffs $n^2$, $n$)
| B1, M1, A1 [3] | co; Correct method. co; [M1 only when coeffs compared]
**(b)** $\frac{a}{1-r} = 20, \frac{a(1-r)^2}{1-r}$, or $a + ar = 12.8$
Elimination of $\frac{a}{1-r}$ or $a$ or $r$
$\rightarrow (r = 0.6) \rightarrow a = 8$ | B1, B1, M1, DM1, A1 [5] | co co; 'Correct' elimination to form equation in $a$ or $r$; Complete method leading to $a = $ Condone $a = 8$ and 32\begin{enumerate}[label=(\alph*)]
\item The sum, $S_n$, of the first $n$ terms of an arithmetic progression is given by $S_n = 32n - n^2$. Find the first term and the common difference. [3]
\item A geometric progression in which all the terms are positive has sum to infinity 20. The sum of the first two terms is 12.8. Find the first term of the progression. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q8 [8]}}