Corrective lens prescriptions use negative diopter sphere values for nearsighted patients. Newly nearsighted in both eyes, Carl's first eyeglasses have a sphere value of negative one point two five in each eye. After a year, his new prescription has a sphere value of negative one point five. In his 3rd year wearing glasses, he has a sphere value of negative one point seven five, and a value of negative two in his 4th year. The sphere value is negative two point two five in his 5th year wearing glasses.

Question

Classify the sequence by completing the sentence.

The following sentence contains two pieces of missing information. You will complete the sentences by selecting an option from a radio group.
Because the terms of the sequence
have a common difference
, the sequence is
arithmetic
.

The following sentence contains a missing piece of information. You will complete the sentence by entering an answer in a text field.
The first term is -1.25. Each term is equal to the previous term plus a common difference. The common difference is

.

Answer 1

To determine if the sequence is arithmetic, we need to check the common difference between the terms.

The sequence is as follows:

• 1st year: -1.25
• 2nd year: -1.50
• 3rd year: -1.75
• 4th year: -2.00
• 5th year: -2.25

Now let's find the common differences between subsequent terms:

• From -1.25 to -1.50:
  \(-1.50 - (-1.25) = -1.50 + 1.25 = -0.25\)

• From -1.50 to -1.75:
  \(-1.75 - (-1.50) = -1.75 + 1.50 = -0.25\)

• From -1.75 to -2.00:
  \(-2.00 - (-1.75) = -2.00 + 1.75 = -0.25\)

• From -2.00 to -2.25:
  \(-2.25 - (-2.00) = -2.25 + 2.00 = -0.25\)

Since the difference between each pair of consecutive terms is consistently -0.25, we conclude that the sequence has a common difference.

Therefore, you can fill in the sentences as follows:

1. Because the terms of the sequence have a common difference, the sequence is arithmetic.
2. The common difference is -0.25.

So the completed sentences are:

1. Because the terms of the sequence have a common difference, the sequence is arithmetic.
2. The common difference is -0.25.