Answer:
Word Problem:
Gracie, Mary, and Nancy are participating in a running challenge. Each day, they run a certain distance, and the distance they run increases or decreases based on a geometric sequence. After a few days, the distances they run form an arithmetic sequence.
On the first day, Gracie runs 2 kilometers, Mary runs 4 kilometers, and Nancy runs 8 kilometers. Each subsequent day, the distance they run is calculated by multiplying the previous day’s distance by a common ratio.
After a few days, they notice that the distances they run form an arithmetic sequence. On the nth day, the distance they run is given by the formula:
Distance = a + (n - 1) × d
Where:
a is the initial distance they run on the first day.
n is the day number.
d is the common difference between the distances they run each day.
To find the common ratio of the geometric sequence, we can divide the distance on the second day by the distance on the first day:
Common Ratio = Distance on Day 2 / Distance on Day 1
Common Ratio = 4 kilometers / 2 kilometers = 2
Now, let’s find the common difference of the arithmetic sequence. We know that the distance on the first day is 2 kilometers, and the common ratio is 2. Using the formula for the nth term of an arithmetic sequence, we can calculate the common difference:
Common Difference = Distance on Day 2 - Distance on Day 1
Common Difference = 4 kilometers - 2 kilometers = 2 kilometers
Therefore, the word problem that starts with a three-number geometric sequence and transforms into an arithmetic sequence is as follows:
Gracie, Mary, and Nancy are participating in a running challenge. On the first day, Gracie runs 2 kilometers, Mary runs 4 kilometers, and Nancy runs 8 kilometers. Each subsequent day, the distance they run is calculated by multiplying the previous day’s distance by 2. After a few days, they notice that the distances they run form an arithmetic sequence, with a common difference of 2 kilometers.