2. The neck of a guitar is divided by frets in such a way that pressing down on

Question

2. The neck of a guitar is divided by frets in such a way that pressing down on each fret changes the note produced when the guitar is played. The first fret of a guitar is placed 36.35 mm from the end of the guitar's neck. The second fret is placed 34.31 mm from the first fret. The distances, d, in milli meters, of the first four frets relative to the previous fret are shown in the graph below. (F- LE.A.2) 45T 42 39 36 (2,34.31) (3, 32.39) (4 30.57) 30 T 27 24 T 21 18 T 15 12 9 6 3 1 2 3 4 5 6 78 9 a. Consider the sequence of distances between the frets. Is the sequence arithmetic or ge common difierence orailo io jusiiiy your answer. b. Write an explicit rule for d(n), the distance between fret n and the fret below it. Show you c. Use your rule from part b to determine the distance between the 19h and 20h frets.

Explanation / Answer

a)

34.31-36.35=-2.04

32.39-34.31=-1.92

=>there is no common difference

34.31/36.3532.39/34.31 30.57/32.39

so the sequence of distances is geometric , common ratio is 34.31/36.350.9438789546

b)

d(1)=36.35,d(2)=36.35*(34.31/36.35),d(3)=36.35*(34.31/36.35)2.................

=>d(n)=36.35*(34.31/36.35)n-1

c)

distance between 19th and 20th frets =d(20)

distance between 19th and 20th frets =36.35*(34.31/36.35)20-1

distance between 19th and 20th frets =12.13 mm approximately

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