| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 5 |
| Topic | Geometric Sequences and Series |
| Type | Form and solve quadratic in parameter |
| Difficulty | Standard +0.8 This question requires understanding of both arithmetic and geometric progressions, forming and solving a quadratic equation from the GP condition, applying convergence criteria (|r| < 1), and calculating sum to infinity. The multi-step nature, need to handle the parameter x systematically, and the convergence constraint make this moderately challenging, though the individual techniques are standard A-level content. |
| 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 |
**(i)** Let $d$ be the common difference, so $b = a + d$ and $c = a + 2d$: M1
Then $a + c = 2a + 2d$
$= 2(a + d) = 2b$ **AG** A1 **[2]**
**(ii)** If $\alpha, \beta, \gamma$ are in GP then $\beta = \alpha r$, $\gamma = \alpha r^2$: M1
So $\alpha\gamma = \alpha^2 r^2 = \beta^2$ A1 **[2]**
**(iii)** $(2 - 3x)(3 - 2x) = (2x)^2$ or equivalent: M1
$\Rightarrow x = 6$ or $0.5$
$\Rightarrow$ common ratio $= -3/4$ or $2$ A1
The first gives a convergent progression (or their $r = 2$ is rejected) A1
with sum: M1
$-64/7$ A1 **[5]**6 (i) Given that the numbers $a , b$ and $c$ are in arithmetic progression, show that $a + c = 2 b$.\\
(ii) Find an analogous result for three numbers in geometric progression.\\
(iii) The numbers $2 - 3 x , 2 x , 3 - 2 x$ are the first three terms of a convergent geometric progression. Find $x$ and hence calculate the sum to infinity.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q6 [5]}}