Question

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?

Answer 1

To develop an explicit formula for the positions of the A keys on a piano, let's first identify the positions of the A keys:

1. The first A (lowest A) is at position 1.
2. The next A is at position 1 + 7 = 8.
3. The following A is at position 8 + 7 = 15.
4. Continuing this pattern, the positions can be expressed as:

- Position of A \( n \): \( a_n = 1 + (n - 1) \cdot 7 \)
- This simplifies to: \( a_n = 1 + 7n - 7 = 7n - 6 \)

This formula represents the position of each A key on the piano, where \( n \) is the \( n^{th} \) A key.

Now, we want to determine the highest A key position on a piano with 52 white keys. To find this, we need to solve for \( n \) in the equation \( a_n \leq 52 \):

\[
7n - 6 \leq 52
\]
Adding 6 to both sides:
\[
7n \leq 58
\]
Dividing by 7:
\[
n \leq \frac{58}{7} \approx 8.2857
\]

Since \( n \) must be a whole number, the maximum value for \( n \) is 8.

Now, we can find the position of the 8th A:
\[
a_8 = 7(8) - 6 = 56 - 6 = 50
\]

Thus, the highest A key on a piano with 52 white keys is in position **50**.