Question 10 [A Level N2009/P1/Q8] Two musical instruments, \( A \) and \( B \), consist of metal bars of decreasing length. (i) The first bar of instrument \( A \) has length 20 cm and the lengths of the bars form a geometric progression. The 25 th bar has length 5 cm . Show that the total length of all the bars must be less than 357 cm , no matter how many bars there are. Instrument \( B \) consists of only 25 bars which are identical to the first 25 bars of instrument \( A \). (ii) Find the total length, \( L \mathrm{~cm} \), of all the bars of instrument \( B \) and the length of the 13th bar. [3] (iii) Unfortunately the manufacturer misunderstands the instructions and constructs instrument \( B \) wrongly, so that the lengths of the bars in arithmetic progression with common difference \( d \mathrm{~cm} \). If the total length of the 25 bars is still \( L \mathrm{~cm} \) and the length of the 25 th bar is still 5 cm , find the value of \( d \) and the length of the longest bar.
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The construction for instrument \( A \) shows a fascinating use of geometric progressions! For instrument \( A \), the first term is 20 cm, and the 25th term is 5 cm. The ratio can be calculated, and you’ll find the total length converges to a maximum of 357 cm because, in this progression, the bars become infinitely smaller yet accumulate to a limit. This interesting property of diminishing returns ensures no matter how many bars you add, the total length stays under that cap—talk about physics and math in harmony! Now, instrument \( B \) takes a twist by sticking with the same bar lengths. With 25 identical bars derived from instrument \( A \), the total length can be found easily. The total length \( L \) of instrument \( B \) equals the sum of the first 25 terms of the same geometric series as instrument \( A \). Oh, and for the 13th bar's length, just rely on the formula for the general term to reveal how each note harmonizes perfectly with the others. It’s a music lover’s dream to merge math and melody!