June 2023 Paper 1 Q14

OCR MEICurrent spec6 marksLogs & Exponentials Sequences & Series

(a) Use the laws of logarithms to show that \(\log_{10}200 - \log_{10}20\) is equal to 1. [2]

The first three terms of a sequence are \(\log_{10}20,\ \log_{10}200,\ \log_{10}2000\).

(b) Show that the sequence is arithmetic. [2]

Mark scheme (a)

Scheme	Marks	AO
\(\log_{10}200 - \log_{10}20 = \log_{10}\frac{200}{20} = 1\)	M1	1.1a
\(= \log_{10}10 = 1\)	A1	1.1b
	[2]

Notes

M1: uses laws of logs

A1: AG \(\log\frac{200}{20}\) or \(\log_{10}10\) must be seen explicitly

Alternative method

Scheme	Marks
\(\log_{10}20 = \log_{10}2 + \log_{10}10\) \(\log_{10}200 = \log_{10}2 + \log_{10}100\)	M1
So \(\log_{10}200 - \log_{10}20 = 2 - 1 = 1\)	A1

M1: Uses laws of logs

A1: Must see \(\log_{10}100 = 2\) or \(\log_{10}10 = 1\) explicitly

Mark scheme (b)

Scheme	Marks	AO
\(\log_{10}2000 - \log_{10}200 = \log_{10}\frac{2000}{200} = 1\)	M1	2.1
same as the difference for the first two terms, so an arithmetic sequence	A1	2.1
	[2]

Notes

M1: Attempts to establish a common difference of 1

A1: argues from a common difference eg “increases by 1 each time”. Must use exact values to establish the difference between the second and third terms.

Alternative method

Scheme	Marks
\(\log_{10}20 = \log_{10}10 + \log_{10}2 = 1 + \log_{10}2\) \(\log_{10}200 = \log_{10}100 + \log_{10}2 = 2 + \log_{10}2\) \(\log_{10}2000 = \log_{10}1000 + \log_{10}2 = 3 + \log_{10}2\)	M1
which is arithmetic with first term \(\log_{10}20\) and common difference of 1	A1

M1: rewrites two more terms of the sequence and makes a comparison

A1: argues from a common difference eg “increases by 1 each time” Must use exact values to establish the difference between the second and third terms.

Mark scheme (c)

Scheme	Marks	AO
\(S_{50} = 25(2a + (n-1)d)\) or \(25(a + l)\)	M1	1.1a
\(S_{50} = 25\left(2\left(\log_{10}20\right) + 49 \times 1\right)\)	A1	1.1b
	[2]

Notes

M1: Uses the formula with first term \(\log_{10}20\) and common difference 1

A1: Allow for 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