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Question: A set of \(56\) tuning forks are so arranged in series that each fork gives \(4\) beats per second w...

A set of 5656 tuning forks are so arranged in series that each fork gives 44 beats per second with the previous one. The frequency of the last fork is 33 times that of the first. The frequency of the first fork is
A.110110
B.6060
C.5656
D.5252

Explanation

Solution

If we observe carefully then the question says that the beats per second of the tuning forks are such that they form an arithmetic progression because, with a common difference of 44 between all the terms. The formula for nthn\text{th} of the arithmetic progression can be used to solve this question.

Complete answer:
Let us assume that the frequency of the first fork is ff, then using our knowledge of arithmetic progression we can write down the frequency of the last fork as follows:
f′=f+(56−1)×4 ⇒f′=f+55×4 ⇒f′=f+220 \begin{aligned} & f'=f+\left( 56-1 \right)\times 4 \\\ & \Rightarrow f'=f+55\times 4 \\\ & \Rightarrow f'=f+220 \\\ \end{aligned}
As it is given that the frequency of the last fork is 33 times that of the first, so we will now put the value of the frequency of the last tuning fork as thrice the frequency of the first tuning fork. The expression will now look like this:
3f=f+220 ⇒3f−f=220 ⇒2f=220 ∴f=110 \begin{aligned} & 3f=f+220 \\\ & \Rightarrow 3f-f=220 \\\ & \Rightarrow 2f=220 \\\ & \therefore f=110 \\\ \end{aligned}
The frequency of the first tuning fork came out to be 110110.

Hence the correct option will be AA.

Additional information:
If some numbers are given in arithmetic progression, let us assume that the first term of the given arithmetic progression is aa, the number of terms given is nn and the common difference between the terms of the given arithmetic progression is dd, then the nthn\text{th}term of the arithmetic progression is given by:
an=a+(n−1)d{{a}_{n}}=a+\left( n-1 \right)d
This same formula has been used to find out the frequency of the last tuning fork.

Note:
As the question says that each of the 5656 tuning forks give 44 beats per second with the previous one, it means that each tuning fork gives a frequency of 44 beats per second more as compared to the previous tuning fork and all of them hence form an arithmetic progression.