| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2019 |
| Session | Specimen |
| Marks | 7 |
| Topic | Geometric Sequences and Series |
| Type | Shared terms between AP and GP |
| Difficulty | Standard +0.8 This question requires setting up simultaneous equations from the AP-GP relationship, algebraic manipulation to find d in terms of a, then solving a quadratic to find r=5/8. The sum to infinity requires expressing the first term in terms of a. It's above average difficulty due to the multi-step algebraic reasoning and the need to connect two sequence types, but follows a recognizable pattern for this topic type. |
| 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 |
(a) State $n$th term of an AP for at least one term. $(a,\ a+8d$ and $a+13d)$ — **M1**
Equate to $ar$ and $ar^2$ $(a+8d = ar,\ a+13d = ar^2)$ — **A1**
State an expression for $r$, $d$ or $r^2$ — **B1**
Equate 2 expressions and make at least one step to solve — **M1**
Obtain an expression for $d$ or $a$: $d = \dfrac{-3a}{64}$ — **A1**
Substitute their value for $d$ or $a$ to find $r$ — **M1**
Obtain $r = \dfrac{5}{8}$ **AG** — **A1**
**Total: 7**
(b) Substitute $r$ into correct formula — **M1**
Obtain $S = \dfrac{8a}{3}$ — **A1**
**Total: 2**11 An arithmetic progression has first term $a$ and common difference $d$. The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio $r$, where $r \neq 1$.
\begin{enumerate}[label=(\alph*)]
\item Find $d$ in terms of $a$ and show that $r = \frac { 5 } { 8 }$.
\item Find the sum to infinity of the geometric progression in terms of $a$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2019 Q11 [7]}}