| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Topic | Geometric Sequences and Series |
| Type | Find year when threshold exceeded |
| Difficulty | Standard +0.3 This is a straightforward geometric progression question requiring standard formulas. Part (i) involves setting up and solving a simple equation (leading to a quadratic that factors nicely). Parts (ii) and (iii) are direct applications of GP formulas with no conceptual challenges—just substitution and logarithm manipulation. Slightly easier than average due to the routine nature of all steps. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(5(a+ar) = a + ar + ar^2 + ar^3\) OR \(\frac{5a(1-r^2)}{1-r} = \frac{a(1-r^4)}{1-r}\) | B1 | |
| Attempt to simplify their sum expression – one step enough | M1 | |
| Obtain \(r = 2\) only from NIS | A1 | [3] |
| (ii) \(a = 12\) | B1 | [1] |
| (iii) Use sum formula for GP with their \(a (= 12), r = 2\) | M1 | |
| Use log law on equation or inequality of the form \(2^n = k\) | Dep M1 | |
| Obtain 209.02 and hence 210 only | A1 | [3] |
(i) $5(a+ar) = a + ar + ar^2 + ar^3$ OR $\frac{5a(1-r^2)}{1-r} = \frac{a(1-r^4)}{1-r}$ | B1 |
Attempt to simplify their sum expression – one step enough | M1 |
Obtain $r = 2$ only from NIS | A1 | [3]
(ii) $a = 12$ | B1 | [1]
(iii) Use sum formula for GP with their $a (= 12), r = 2$ | M1 |
Use log law on equation or inequality of the form $2^n = k$ | Dep M1 |
Obtain 209.02 and hence 210 only | A1 | [3]A geometric progression with common ratio $r$ consists of positive terms. The sum of the first four terms is five times the sum of the first two terms.
\begin{enumerate}[label=(\roman*)]
\item Find an equation in $r$ and deduce that $r = 2$. [3]
\item Given that the fifth term is 192, find the value of the first term. [1]
\item Find the smallest value of $n$ such that the sum of the first $n$ terms of the progression exceeds $10^{64}$. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2010 Q6 [7]}}