Question

Leonardo has $525 in a savings account. After one month, he has $580 in the account. The next month, the balance is $635. The balance after the third month is $690.

Write the sequence represented by the story.
Is the sequence arithmetic or geometric?
Does it have a common difference or common ratio?
What is the common difference or common ratio?
What are the next three terms?

Answer 1

Let's define the sequence based on the balances in Leonardo's savings account at the end of each month:

1. After the first month: $580
2. After the second month: $635
3. After the third month: $690

Thus, the sequence starts from $525 and goes as follows:
- First term: 525
- Second term: 580
- Third term: 635
- Fourth term: 690

The sequence can be represented as: \( 525, 580, 635, 690 \)

### Is the sequence arithmetic or geometric?
The sequence is arithmetic because the difference between consecutive terms is constant.

### Common difference or common ratio?
In this case, since the sequence is arithmetic, we have a common difference.

### What is the common difference?
To find the common difference, we can subtract the first term from the second term:

- \( 580 - 525 = 55 \)
- \( 635 - 580 = 55 \)
- \( 690 - 635 = 55 \)

So, the common difference is \( 55 \).

### Next three terms?
To find the next three terms in the sequence, we can continue adding the common difference of \( 55 \):

1. Fifth term: \( 690 + 55 = 745 \)
2. Sixth term: \( 745 + 55 = 800 \)
3. Seventh term: \( 800 + 55 = 855 \)

Thus, the next three terms are \( 745, 800, 855 \).

### Summary
- Sequence: \( 525, 580, 635, 690 \)
- Type: Arithmetic
- Common difference: \( 55 \)
- Next three terms: \( 745, 800, 855 \)