There are \[26\] tuning forks arranged in the decreasing order of their frequencies. Each tuning fork gives \[3\] beats with the next. The first one is octave of the last. What is the frequency of \[{18^{{\text{th}}}}\] tuning fork?
A. \[100\;{\text{Hz}}\]
B. \[99\;{\text{Hz}}\]
C. \[96\;{\text{Hz}}\]
D. \[103\;{\text{Hz}}\]

Hint: Assume the frequency for the last fork as \[N\] and obtain the same for the first fork according to the question. Make a sequence to obtain the frequency for each and every fork number and finally deduce the sequence for the last fork. Compare it with \[N\] and calculate it.

Complete step by step solution: Given, total number of tuning forks \[ = 26\]
Let us assume the frequency of the last fork is \[N\].
According to the question, the first one is octave of the last.
Therefore, the frequency of the first tuning fork is \[2N\].

Since, each tuning fork gives \[3\] beats with the next.
Therefore, the frequency of the second tuning fork
\[ = 2N - 3\].
Similarly, the frequency of the third tuning fork
\[ = 2N - 2 \times 3\].
The frequency of the fourth tuning fork
\[ = 2N - 3 \times 3\].

From this observation we can see that there becomes a sequence of
\[2N - (N - 1) \times 3\]

Hence, the frequency of the \[{26^{{\text{th}}}}\]tuning fork
\[ = 2N - 25 \times 3\].

Now,
\[N = 2N - 25 \times 3\]
Solve the above equation to obtain the value of \[N\].
Therefore,
\[
  N = 25 \times 3 \\
  N = 75 \\
 \]

Hence, the frequency of \[{26^{{\text{th}}}}\] wave is \[75\;{\text{Hz}}\].

Now, the frequency of \[{\text{1}}{{\text{8}}^{{\text{th}}}}\] wave is given by, \[{\text{2}}N - 17 \times 3\]
Place the value of \[N = 75\] in the above equation.

Therefore, the frequency of \[{\text{1}}{{\text{8}}^{{\text{th}}}}\] wave is given is
\[
   = \left( {{\text{2}} \times 75} \right) - \left( {17 \times 3} \right) \\
   = 150 - 51 \\
   = 99\;{\text{Hz}} \\
 \]

Hence, option B is correct.

Note: In this problem we have to find the frequency for \[{\text{1}}{{\text{8}}^{{\text{th}}}}\] fork tuning. In order to find it we need to calculate the frequency of the first fork. According to the question, the first one is octave of the last. It means the first one is the double of the last. Don’t get confused with the last one is the double of the first one.