| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find N for S_∞ - S_N condition |
| Difficulty | Standard +0.3 This is a straightforward geometric sequence problem requiring standard formulas. Part (a) involves dividing two terms to find r (routine manipulation), part (b) uses substitution to find a, and part (c) requires solving S_n > 156 using the sum formula—all standard textbook techniques with no novel insight needed. Slightly easier than average due to the guided structure. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(r = \sqrt{\frac{12.8}{20}} = 0.8\) or \(20 \times 0.8 \times 0.8 = 12.8\) so \(r = 0.8\) | B1* | Correctly demonstrates \(r = 0.8\) with no incorrect working; minimum acceptable is an expression for \(r\) not 0.8 or equation involving \(r\) with all values substituted; \(r\) must be linked with 0.8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a = 20 \div 0.8^2\) | M1 | Correct method for first term; score for \(12.8 \div 0.8^4\) or \(20 \div 0.8^2\); work seen in (a) must be used or stated |
| \(= 31.25\) | A1 | Accept \(\frac{125}{4}\) or \(31\frac{1}{4}\) with no incorrect working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{31.25(1 - 0.8^n)}{1 - 0.8} > 156\) | M1 | Attempts to set up equation/inequality using first term and \(r = 0.8\) in sum formula; allow any inequality or \(=\) |
| \(1 - 0.8^n > 0.9984 \Rightarrow 0.8^n < 0.0016\) | dM1 | Rearranges to \(A \times 0.8^n \ldots B\); dependent on first M1; may be implied by correct logarithm work |
| \(0.8^n < 0.0016 \Rightarrow n > \frac{\log(0.0016)}{\log(0.8)}\) or \(n > \log_{0.8} 0.0016\) | M1 | Attempts to find \(n\) by solving \(0.8^n \ldots 0.0016\) using logarithms correctly; allow any inequality or \(=\) |
| \(n = 29\) | A1 | cso; all three M marks must have been earned; must come from correct equation (allow any inequality) |
## Question 7:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $r = \sqrt{\frac{12.8}{20}} = 0.8$ or $20 \times 0.8 \times 0.8 = 12.8$ so $r = 0.8$ | B1* | Correctly demonstrates $r = 0.8$ with no incorrect working; minimum acceptable is an expression for $r$ not 0.8 or equation involving $r$ with all values substituted; $r$ must be linked with 0.8 |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $a = 20 \div 0.8^2$ | M1 | Correct method for first term; score for $12.8 \div 0.8^4$ or $20 \div 0.8^2$; work seen in (a) must be used or stated |
| $= 31.25$ | A1 | Accept $\frac{125}{4}$ or $31\frac{1}{4}$ with no incorrect working |
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{31.25(1 - 0.8^n)}{1 - 0.8} > 156$ | M1 | Attempts to set up equation/inequality using first term and $r = 0.8$ in sum formula; allow any inequality or $=$ |
| $1 - 0.8^n > 0.9984 \Rightarrow 0.8^n < 0.0016$ | dM1 | Rearranges to $A \times 0.8^n \ldots B$; dependent on first M1; may be implied by correct logarithm work |
| $0.8^n < 0.0016 \Rightarrow n > \frac{\log(0.0016)}{\log(0.8)}$ or $n > \log_{0.8} 0.0016$ | M1 | Attempts to find $n$ by solving $0.8^n \ldots 0.0016$ using logarithms correctly; allow any inequality or $=$ |
| $n = 29$ | A1 | cso; all three M marks must have been earned; must come from correct equation (allow any inequality) |\begin{enumerate}
\item A geometric sequence has first term $a$ and common ratio $r$, where $r > 0$
\end{enumerate}
Given that
\begin{itemize}
\item the 3rd term is 20
\item the 5th term is 12.8\\
(a) show that $r = 0.8$\\
(b) Hence find the value of $a$.
\end{itemize}
Given that the sum of the first $n$ terms of this sequence is greater than 156\\
(c) find the smallest possible value of $n$.\\
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\hfill \mbox{\textit{Edexcel P2 2023 Q7 [7]}}