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28 Sep 2024, 06:27
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Given: A certain musical scale has has 13 notes, each having a different frequency, measured in cycles per second. In the scale, the notes are ordered by increasing frequency, and the highest frequency is twice the lowest. For each of the 12 lower frequencies, the ratio of a frequency to the next higher frequency is a fixed constant.
Asked: If the lowest frequency is 440 cycles per second, then the frequency of the 7th note in the scale is how many cycles per second?
The lowest frequency = 440 cycles per second
The highest frequency = 2*440 = 880 cycles per second
Since the frequency have a common ratio, they form a geometric progression with : -
440, 440r, 440r^2,..., 440r^12=880
a = 440
l = 880
\(l = 880 = ar^{n-1} = 440r^{12}\)
\(r^{12} = 2 \)
\(r= 2^{1/12}\)
The frequency of the 7th note in the scale = \(ar^{7-1} = ar^6 = 440*2^{6/12} = 440*2^{1/2} = 440*\sqrt2\)
IMO A
Given: A certain musical scale has has 13 notes, each having a different frequency, measured in cycles per second. In the scale, the notes are ordered by increasing frequency, and the highest frequency is twice the lowest. For each of the 12 lower frequencies, the ratio of a frequency to the next higher frequency is a fixed constant.
Asked: If the lowest frequency is 440 cycles per second, then the frequency of the 7th note in the scale is how many cycles per second?
The lowest frequency = 440 cycles per second
The highest frequency = 2*440 = 880 cycles per second
Since the frequency have a common ratio, they form a geometric progression with : -
440, 440r, 440r^2,..., 440r^12=880
a = 440
l = 880
\(l = 880 = ar^{n-1} = 440r^{12}\)
\(r^{12} = 2 \)
\(r= 2^{1/12}\)
The frequency of the 7th note in the scale = \(ar^{7-1} = ar^6 = 440*2^{6/12} = 440*2^{1/2} = 440*\sqrt2\)
IMO A
29 Sep 2024, 02:55
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The ratio of (n+1)th term and nth term is constant. This is GP
nth term can be written as
nth term = first term x common ratio^n-1
we know the 13th term = 880 and first term = 440
we can write
13th term = first term x common ratio^n-1
880 = 440 x r^12
r = 2^1/12
now
7th term = 440 x (2^1/12)^6 = 440 x 2^1/2
hence A
nth term can be written as
nth term = first term x common ratio^n-1
we know the 13th term = 880 and first term = 440
we can write
13th term = first term x common ratio^n-1
880 = 440 x r^12
r = 2^1/12
now
7th term = 440 x (2^1/12)^6 = 440 x 2^1/2
hence A
