Question

Make Sense and Persevere The lowest and leftmost note on a piano keyboard is an A. The next lowest A is seven white keys to the right. This pattern continues. Write an explicit formula for an arithmetic sequence to represent the position of each A key on the piano, counting from the left. If a piano has 52 white keys, in what position is the key that plays the highest A?

a) Position formula: A_n = 7n - 5; Position of highest A: A_8 = 53
b) Position formula: A_n = 7n - 4; Position of highest A: A_7 = 48
c) Position formula: A_n = 7n - 6; Position of highest A: A_9 = 59
d) Position formula: A_n = 7n - 7; Position of highest A: A_10 = 62

Answer 1

The explicit formula for the position of each A key on a piano is An = 7n - 6. For a piano with 52 white keys, the position of the highest A is calculated by setting 7n - 6 = 52 and solving for n, giving us the position A8 = 50.

Step-by-step explanation:

We want to determine the explicit formula for the position of each A key on a piano keyboard. The first A is the leftmost key, and then every A after that is seven white keys to the right. So, we start with the first A at position 1, the second A at position 8, the third A at position 15, and so on.

An arithmetic sequence can be defined in the form An = d(n-1) + A1, where d is the common difference, n is the term number, and A1 is the first term in the sequence. For this sequence, our first term A1 is 1 (the first A) and the common difference d is 7 since each A is seven white keys to the right of the previous one.

Therefore, the explicit formula for the position of each A key is An = 7(n-1) + 1. Simplifying, we get An = 7n - 6. If there are 52 white keys, then we set 7n - 6 = 52 and solve for n to find the position of the highest A.

Solving 7n - 6 = 52 gives us 7n = 58, and n = 58/7, which simplifies to n = 8.28, which doesn't make sense in this context as n must be an integer. Since n is not a whole number, we find the nearest whole number which is n = 8. Substituting n = 8 into our formula gives A8 = 7(8) - 6 = 56 - 6 = 50. Therefore, the key that plays the highest A has the position A8 = 50.