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The information discusses the frequencies of different notes on a piano and the patterns they form. It explains how to find the common ratio in a geometric series and how to calculate the frequency of a specific note. It also discusses the highest frequency a human can hear and how to determine the frequency and number of the highest A note. Additionally, it covers a profit formula and how to find the price and quantity that maximize profit. The maximum profit is stated to be 1050. The first question is, the lowest and leftmost note on a piano is called A0 and has a frequency of 27.5 Hz. The note A1, one octave above A0, has a frequency of 55 Hz, and the note A2 has a frequency of 110 Hz. So for part A, we have been asked to write down the pattern of frequencies and determine what kind of sequence they form. So at first, let us see the differences between two of the frequencies, 27.5 and 55. We can notice that the difference is not equal to the difference between 110 and 55. So we can understand from that that it's a geometric series. So the way we are going to prove it is, at first, we are going to consider A0 to be A, which is our first term, or T0, and then A1 to be T1, which has the formula AR, and A2 to be term 2, which has the formula AR2. So to find the common ratio, we have to divide T2 by T1, and we got the common ratio to be 2. For the second part of the question, we have been told to find the frequency of the note A10. So the formula for that will be A into R to the power 10. And after inserting the values, we get that the answer is 28160. For the third part, we have been said the highest frequency a human can hear is approximately 20,000. Determine the frequency and the number of the highest A note a human can hear. So we have been given the sum, which is 20,000, and the formula for the sum in geometric progression is AR to the power n minus 1 divided by R minus 1. So we're going to insert all of our values. And in the third step, we have cross-multiplied 27.5 times 2 to the power n minus 27.5. We multiply the whole thing. We actually cross-multiply 20,000 with 1. And then we got 27.5 into 2 to the power n minus 27.5 equals 20,000. And then after solving, we have to start solving for n. So we separated 2 to the power n and brought everything to the right side. And then we got the equation 2 to the power n equals to 811 by 11. And then we are going to turn it into a logarithm form. So n equals to log base 2, 811 by 11. And then we got the value of n to be 9.5, which is 9. So the number is 9. And now we have to find what T9 is. So the formula for T9 will be A into R to the power 9. And by inserting the values, we got the answer to be 14080 Hertz. For question number 3, we have been given a formula for P, which is a profit, 100 minus 0.5Q, where P is the price. P represents the price, and Q is the quantity. Then in part A, we have been given a profit formula, which is CQ minus 3,000 minus 10Q. And what price should you pay the company charge per toy to make the amount of money? So we have been asked to find the cost, which is the P, and to make the most amount of money. So the first equation we have, P, I have taken it to be the equation 1. Then I have inserted that value of P, equation 1, into the second formula, which is the formula for the profit. And then I have found an equation where we only have Q. And then I have differentiated it to find the value of Q to be 90, and then found the value of P, for which the profit is going to be the maximum profit. And for the B part, they have asked what the maximum profit is and for what quantity it is achieved. So we find the value of Q, and then we insert the value of P and Q into the profit formula. And we get the answer to be 1, 0, 5, 0, which is the profit.