Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

A bicycle rider coasts downhill, traveling 4 feet the first second. In each succeeding second, the rider travels 5 feet farther than in the preceding second. If the rider reaches the bottom of the hill in 11 seconds, find the total distance traveled.

Step-by-Step Solution

Answer

The total distance traveled is 319 feet.

Step 1: Understanding the Problem

The bicycle rider starts by traveling 4 feet in the first second. Each subsequent second, the distance increases by 5 feet more than the previous second. This continues for 11 seconds.

Step 2: Formulate the Sequence

The distance traveled each second forms an arithmetic sequence: 4, 9, 14, ..., and so on. We can identify the first term as 4 and the common difference as 5.

Step 3: Determine the n-th Term Formula

The formula for the n-th term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1)d \]where \( a_1 = 4 \) and \( d = 5 \).

Step 4: Calculate the Last Term

Substitute \( n = 11 \) into the formula to find the last term:\[ a_{11} = 4 + (11 - 1) imes 5 = 4 + 50 = 54 \]Thus, the distance traveled in the 11th second is 54 feet.

Step 5: Sum the Sequence

The sum of an arithmetic sequence is found using the formula:\[ S_n = rac{n}{2} (a_1 + a_n) \]Substitute \( n = 11 \), \( a_1 = 4 \), and \( a_{11} = 54 \):\[ S_{11} = rac{11}{2} (4 + 54) = rac{11}{2} imes 58 = 319 \]The total distance traveled is 319 feet.

Key Concepts

Understanding the Sum of an Arithmetic Sequence

When dealing with arithmetic sequences, one important concept is determining the sum. The sum of an arithmetic sequence involves adding up all the terms. In our example, the sequence began with 4 feet in the first second and increased by 5 feet each subsequent second. This forms a sequence of numbers such as 4, 9, 14, and so on, up to the 11th term.
To find the sum of such a sequence, you can use the formula: \[ S_n = \frac{n}{2} (a_1 + a_n) \]

• \( S_n \) stands for the sum of the first \( n \) terms of the sequence.
• The number \( n \) is the number of terms you want to sum, which is 11 in this case.
• \( a_1 \) is the first term of the sequence.
• \( a_n \) is the n-th term of the sequence.

Using this formula, we've calculated that the bicycle rider traveled a total distance of 319 feet in 11 seconds. This formula provides a quick way to find the total distance without adding each term separately.

Identifying the Common Difference

The common difference is an essential component of any arithmetic sequence. It represents the constant amount by which each term increases over the previous one.
In the bicycle rider's example, we saw that each second, the distance increased by 5 feet. This means our common difference, denoted as \( d \), is 5.
Recognizing the common difference allows you to easily generate the terms of the sequence after the first one. For instance, starting from the initial term of 4 feet, adding this common difference repeatedly constructs the sequence: 4, 9, 14, ..., each separated by an increment of 5 feet.
Understanding the common difference helps not only in generating terms but also in using it effectively to apply formulas related to arithmetic sequences.

Applying the n-th Term Formula in Arithmetic Sequences

To find a specific term, known as the n-th term, in an arithmetic sequence, we use a simple formula that incorporates the first term and the common difference. This formula is written as:\[ a_n = a_1 + (n - 1) \times d \]

• \( a_n \) is the n-th term you are trying to find.
• \( a_1 \) is the first term of the sequence, and in our case, it is 4.
• \( n \) is the term position in the sequence. For the 11th term, it's 11.
• \( d \) is the common difference.

Using this formula, we calculated that the 11th term of our sequence is 54 feet. This formula is handy when you need to find a specific term without listing all prior terms.

Exploring Distance Calculation in Sequences

Distance calculation in sequences involves determining how far a moving object like our bicycle rider travels over a period of time when following a predictable pattern of movement. Each second, the rider increases their travel distance by a fixed amount, forming an arithmetic sequence.
To find the total accumulated distance over a set period, we apply the sum of an arithmetic sequence. By summing the individual distances for the 11 seconds as shown in the solution, we find that 319 feet is the total distance traveled. This process demonstrates how arithmetic sequences can be applied in real-world scenarios, like movement over time.
Such calculations are particularly useful in predicting future distances, managing travel plans, or even calculating tasks that occur in a stepwise process over time.