Problem 17 - Algebra for College Students

Question

An object falling from rest in a vacuum falls approximately 16 feet the first second, 48 feet the second second, 80 feet the third second, 112 feet the fourth second, and so on. How far will it fall in 11 seconds? 1936 feet

Step-by-Step Solution

Answer

The object falls 1936 feet in 11 seconds.

Step 1: Understand the Pattern

The distances fallen each second create an arithmetic sequence. Notice the increase in distance: from 16 to 48 is an increase of 32 feet, from 48 to 80 is 32 feet, and so on. This forms an arithmetic series with the first term (s_1) of 16 feet and a common difference (d) of 32 feet.

Step 2: Use the Arithmetic Series Formula

The distance fallen each second can be calculated using the formula for the nth term of an arithmetic sequence: \( s_n = s_1 + (n-1) imes d \). To find the distance fallen in 11 seconds, we need to find the sum of the first 11 terms.

Step 3: Calculate the Distance Each Second

Compute the 11th term using the formula: \( s_{11} = 16 + (11-1) imes 32 = 16 + 320 = 336 \).

Step 4: Sum the Distances for 11 Seconds

Use the formula for the sum of the first n terms of an arithmetic series: \( S_n = \frac{n}{2} (s_1 + s_n) \). For 11 seconds, \( S_{11} = \frac{11}{2} (16 + 336) = \frac{11}{2} \times 352 = 11 \times 176 = 1936 \) feet.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a set of numbers where the difference between any two consecutive terms remains constant. This difference is known as the "common difference."
Each term in an arithmetic sequence can be generated by adding this common difference to the previous term. You start with the first term, also known as the initial term, and proceed step by step by adding the common difference over and over again.
For example, consider an arithmetic sequence with the initial term of 16 and a common difference of 32. The sequence would then become 16, 48, 80, 112, etc. In this sequence, each term increases by 32, giving us a clear pattern to follow.

Common Difference

The common difference is a key component of an arithmetic sequence. It is the value consistently added to each term to obtain the next term in the sequence.
In the exercise, the falling object increased the distance each second by 32 feet. Thus, the common difference is 32. Understanding the common difference is crucial because it allows you to predict future terms in the sequence.

• Identify the first term of your sequence.
• Determine the incremental value added to (or subtracted from) each term.
• Apply this consistent value across the sequence to calculate additional terms.

This regularity makes arithmetic sequences simple to understand and compute.

Nth Term

Finding the nth term of an arithmetic sequence involves a specific formula. You can use this formula to find any term in the sequence, given the first term and the common difference.
The formula is: \[ a_n = a_1 + (n-1) \times d \]Where:

• \( a_n \) is the nth term you're trying to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the term number you want to find.
• \( d \) is the common difference.

You plug in the numbers from your sequence into the formula.
For example, in our sequence where the first term is 16 and the common difference is 32, you find the 11th term by plugging the values into the formula, resulting in 336 feet for the 11th second.

Sum of Series

The sum of an arithmetic series refers to the total of all terms from the first term to the nth term.
The sum of an arithmetic series can be easily calculated using a formula:\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]Where:

• \( S_n \) is the sum of the series.
• \( n \) is the number of terms.
• \( a_1 \) is the first term.
• \( a_n \) is the nth term.

This formula takes the average of the first and the nth term and multiplies it by the number of terms.
For example, the distance the object falls in 11 seconds is found by calculating the sum of the first 11 terms. With 16 feet as the first term and 336 feet as the 11th term, we plug these into the formula to find the sum is 1936 feet, representing the total distance fallen in that time.