# A Level Maths Sequences Question

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#1
Q) A pendulum is set swinging.
Its first oscillation is through an angle of 30 degrees and each succeeding oscillation is through 95% of the angle of the one before.
After how many swings is the angle through which it swings less than 1 degree?

A) My answer was 67 swings, but on the mark scheme, it says '68th swing is less than 1 degree'
Could anyone explain why my answer is wrong and why the correct answer is 68 please? Thanks in advance Last edited by abovethecl0uds; 4 weeks ago
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4 weeks ago
#2
(Original post by abovethecl0uds)
Q) A pendulum is set swinging.
Its first oscillation is through an angle of 30 degrees and each succeeding oscillation is through 95% of the angle of the one before.
After how many swings is the angle through which it swings less than 1 degree?

A) My answer was 67 swings, but on the mark scheme, it says '68th swing is less than 1 degree'
Could anyone explain why my answer is wrong and why the correct answer is 68 please? Thanks in advance The problem is using a base of 0.95 for your logarithms. If you use a base between 0 and 1, then it is no longer true that x < y leads to log(x) < log(y). Instead, we have that x < y leads to log(x) > log(y). You should be able to see why this is the case, but if you just want a demonstration, try plotting y = log(x) for a few different bases in Geogebra.

A clue that things have gone wrong is that you get n < ... , whereas it is obvious that the angle will be less than 1 degree once n has passed a certain point.
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#3
(Original post by Pangol)
The problem is using a base of 0.95 for your logarithms. If you use a base between 0 and 1, then it is no longer true that x < y leads to log(x) < log(y). Instead, we have that x < y leads to log(x) > log(y). You should be able to see why this is the case, but if you just want a demonstration, try plotting y = log(x) for a few different bases in Geogebra.

A clue that things have gone wrong is that you get n < ... , whereas it is obvious that the angle will be less than 1 degree once n has passed a certain point.
I understand that the inequality sign flips when a base between 0 and 1 is used.

If you don't mind answering another question, please could you explain how I get the same answer using base 10?
Last edited by abovethecl0uds; 4 weeks ago
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4 weeks ago
#4
(Original post by abovethecl0uds)
I understand that the inequality sign flips when a base between 0 and 1 is used.

If you don't mind answering another question, please could you explain how I get the same answer using base 10?
Apart from having to be careful with signs, it won't matter which base you use. The individual log values will be different of course, but their ratio will be the same.
1
#5
(Original post by Pangol)
Apart from having to be careful with signs, it won't matter which base you use. The individual log values will be different of course, but their ratio will be the same.
Thank you so much! Everything makes so much sense now 0
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