| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Logarithmic arithmetic progression |
| Difficulty | Moderate -0.8 This question tests basic logarithm laws and arithmetic sequence formulas in a straightforward manner. Part (a) is simple manipulation (log 200 - log 20 = log 10 = 1), part (b) requires showing constant differences using log laws (each term adds log 10 = 1), and part (c) applies the standard arithmetic series formula. All steps are routine applications of well-practiced techniques with no problem-solving insight required, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\log_{10}200 - \log_{10}20 = \log_{10}\frac{200}{20} = 1\) | M1 | 1.1a |
| \(= \log_{10}10 = 1\) | A1 | 1.1b |
| Alternative: \(\log_{10}20 = \log_{10}2 + \log_{10}10\); \(\log_{10}200 = \log_{10}2 + \log_{10}100\) | M1 | |
| So \(\log_{10}200 - \log_{10}20 = 2 - 1 = 1\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\log_{10}2000 - \log_{10}200 = \log_{10}\frac{2000}{200} = 1\) | M1 | 2.1 |
| Same as the difference for the first two terms, so an arithmetic sequence | A1 | 2.1 |
| Alternative: \(\log_{10}20 = \log_{10}10 + \log_{10}2 = 1 + \log_{10}2\); \(\log_{10}200 = \log_{10}100 + \log_{10}2 = 2 + \log_{10}2\); \(\log_{10}2000 = \log_{10}1000 + \log_{10}2 = 3 + \log_{10}2\) | M1 | |
| Which is arithmetic with first term \(\log_{10}20\) and common difference 1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(S_{50} = 25(2a + (n-1)d)\) or \(25(a + l)\) | M1 | 1.1a |
| \(S_{50} = 25\!\left(2\!\left(\log_{10}20\right) + 49 \times 1\right)\) | A1 | 1.1b |
## Question 14(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\log_{10}200 - \log_{10}20 = \log_{10}\frac{200}{20} = 1$ | M1 | 1.1a | Uses laws of logs |
| $= \log_{10}10 = 1$ | A1 | 1.1b | AG $\log\frac{200}{20}$ or $\log_{10}10$ must be seen explicitly |
| **Alternative:** $\log_{10}20 = \log_{10}2 + \log_{10}10$; $\log_{10}200 = \log_{10}2 + \log_{10}100$ | M1 | | Uses laws of logs |
| So $\log_{10}200 - \log_{10}20 = 2 - 1 = 1$ | A1 | | Must see $\log_{10}100 = 2$ or $\log_{10}10 = 1$ explicitly |
**Total: [2]**
## Question 14(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\log_{10}2000 - \log_{10}200 = \log_{10}\frac{2000}{200} = 1$ | M1 | 2.1 | Attempts to establish a common difference of 1 |
| Same as the difference for the first two terms, so an arithmetic sequence | A1 | 2.1 | Argues from a common difference e.g. "increases by 1 each time". Must use exact values to establish the difference between second and third terms |
| **Alternative:** $\log_{10}20 = \log_{10}10 + \log_{10}2 = 1 + \log_{10}2$; $\log_{10}200 = \log_{10}100 + \log_{10}2 = 2 + \log_{10}2$; $\log_{10}2000 = \log_{10}1000 + \log_{10}2 = 3 + \log_{10}2$ | M1 | | Rewrites two more terms of the sequence and makes a comparison |
| Which is arithmetic with first term $\log_{10}20$ and common difference 1 | A1 | | Argues from a common difference. Must use exact values |
**Total: [2]**
## Question 14(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $S_{50} = 25(2a + (n-1)d)$ or $25(a + l)$ | M1 | 1.1a | Uses the formula with first term $\log_{10}20$ and common difference 1 |
| $S_{50} = 25\!\left(2\!\left(\log_{10}20\right) + 49 \times 1\right)$ | A1 | 1.1b | Allow fully correct expression not simplified. ISW. Correct forms include $50\log_{10}20 + 1225$, $25\log 400 + 1225$, $25(\log 400 + 49)$, $25(\log 4 + 51)$ etc |
**Total: [2]**
---14
\begin{enumerate}[label=(\alph*)]
\item Use the laws of logarithms to show that $\log _ { 10 } 200 - \log _ { 10 } 20$ is equal to 1 .
The first three terms of a sequence are $\log _ { 10 } 20 , \log _ { 10 } 200 , \log _ { 10 } 2000$.
\item Show that the sequence is arithmetic.
\item Find the exact value of the sum of the first 50 terms of this sequence.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2023 Q14 [6]}}