Marco went on a hiking trip. The first day he walked 24 kilometers. Each day since, he walked a third of what he walked the day before.
Which expression gives the total distance Marco walked in the first $n$ days of his trip?
Choose 1 answer:
A) $24 \cdot \frac{1-\left(\frac{1}{3}\right)^n}{1-\left(\frac{1}{3}\right)}$
B) $24 \cdot \frac{1-\left(\frac{2}{3}\right)^n}{1-\left(\frac{2}{3}\right)}$
Answer (1)
Identify the first term a = 24 and common ratio r = 3 1 of the geometric sequence.
Recall the formula for the sum of the first n terms of a geometric sequence: S n = a c d o t 1 − r 1 − r n .
Substitute a and r into the formula: S n = 24 c d o t 1 − 3 1 1 − ( 3 1 ) n .
The expression for the total distance is 24 c d o t 1 − ( 3 1 ) 1 − ( 3 1 ) n .
Explanation
Problem Analysis Let's analyze the problem. Marco's hiking distances each day form a geometric sequence. We need to find the expression that represents the total distance he walked in the first n days.
Identifying the Sequence The first day, Marco walked 24 kilometers. So, the first term of the geometric sequence is a = 24 . Each day since, he walked a third of what he walked the day before. This means the common ratio of the geometric sequence is r = 3 1 .
Recalling the Formula The sum of the first n terms of a geometric sequence is given by the formula: S n = a ⋅ 1 − r 1 − r n where S n is the sum of the first n terms, a is the first term, and r is the common ratio.
Substituting Values Now, substitute the values of a and r into the formula: S n = 24 ⋅ 1 − 3 1 1 − ( 3 1 ) n This expression gives the total distance Marco walked in the first n days of his trip.
Final Answer Comparing the derived expression with the given options, we see that it matches option (A).
Examples
Geometric sequences are useful in many real-world scenarios, such as calculating the depreciation of an asset or modeling the spread of a disease. For example, if a car's value decreases by 20% each year, we can use a geometric sequence to determine its value after a certain number of years. Similarly, understanding geometric sequences helps in calculating compound interest, where the interest earned each year is added to the principal, and the next year's interest is calculated on the new amount.
