| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Topic | Geometric Sequences and Series |
| Type | Form and solve quadratic in parameter |
| Difficulty | Moderate -0.3 This is a straightforward geometric progression question requiring standard techniques: forming equations from the GP property (middle term squared equals product of outer terms), solving a quadratic, finding common ratios, and calculating sum to infinity. All steps are routine applications of formulas with no novel insight required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
**(i)** Attempt to use an expression for $r$, e.g. $\frac{6}{x} = \frac{x+5}{6}$ or $\frac{36}{x^2} = \frac{x+5}{x}$ — M1
Obtain correctly $x^2 + 5x - 36 = 0$ AG — A1
Obtain $x = 4$ or $-9$ — B1 **[3]**
**(ii)** Obtain $r = \frac{3}{2}$ — B1
Obtain $r = \frac{-2}{3}$ and only these — B1 **[2]**
**(iii)** State $r = -\frac{2}{3}$ or imply this by considering only this value of $r$ — B1
Attempt to solve $ar^2 = 6$ or $ar = -9$ — M1
Obtain $a = 13.5$ — A1
Use correct sum to infinity formula and obtain $8.1$ — B1 **[4]**
SR both $r$ offered with no choice M1 only9 It is given that $x , 6$ and $x + 5$ are consecutive terms of a geometric progression.\\
(i) Show that $x ^ { 2 } + 5 x - 36 = 0$ and find the possible values of $x$.\\
(ii) Hence find the possible values of the common ratio.
Furthermore, $x , 6$ and $x + 5$ are the second, third and fourth terms of a geometric progression for which the sum to infinity exists.\\
(iii) Find the first term and the sum to infinity.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2014 Q9 [9]}}