| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2016 |
| Session | Specimen |
| Marks | 9 |
| 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 prove r=5/8. Part (ii) applies the sum to infinity formula. It's above average difficulty due to the multi-step algebraic reasoning and the need to connect two sequence types, but follows a fairly standard problem structure for Further Maths students. |
| 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 |
**Question 11(i)**
- State $n$th term of an AP for at least one term $(a,\ a+8d\ \text{and}\ a+13d)$ — M1
- Equate to $ar$ and $ar^2$ $\big(a+8d = ar,\ a+13d = ar^2\big)$ — 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
**Question 11(ii)**
- Substitute $r$ into correct formula — M1
- Obtain $S = \dfrac{8a}{3}$ — A1
**Total: 9 marks**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 $r \neq 1$.\\
(i) Find $d$ in terms of $a$ and show that $r = \frac { 5 } { 8 }$.\\
(ii) Find the sum to infinity of the geometric progression in terms of $a$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2016 Q11 [9]}}