Problem 23 - Algebra for College Students

Question

A pile of logs has 25 logs in the bottom layer, 24 logs in the next layer, 23 logs in the next layer, and so on, until the top layer has 1 log. How many logs are in the pile? 325 logs

Step-by-Step Solution

Answer

The pile has 325 logs.

Step 1: Identify the Sequence

The problem describes a sequence where the number of logs decreases by 1 in each subsequent layer, starting from 25 logs at the bottom layer and ending with 1 log at the top layer. This sequence is an arithmetic sequence.

Step 2: Determine the Terms of the Sequence

The first term (\(a_1\) of the arithmetic sequence is 25 and the last term (\(a_n\) is 1. The common difference (\(d\)) is -1 because each term decreases by 1.

Step 3: Calculate the Number of Terms

To find the total number of terms (\(n\)), use the formula for the last term of an arithmetic sequence: \(a_n = a_1 + (n-1) \cdot d\). Substituting the known values, we have \(1 = 25 + (n-1)(-1)\), which simplifies to \(n = 25\).

Step 4: Find the Sum of the Sequence

Use the formula for the sum of an arithmetic series: \(S_n = \frac{n}{2} \cdot (a_1 + a_n)\). Here, \(n = 25\), \(a_1 = 25\), and \(a_n = 1\). Substitute these values to find the sum: \(S_{25} = \frac{25}{2} \cdot (25 + 1) = 25 \cdot 13 = 325\).

Step 5: Verify the Result

The calculated sum, 325 logs, matches the given total in the problem statement. Hence, the solution is confirmed to be correct.

Key Concepts

Arithmetic Sequence Sum

In an arithmetic sequence, each term increases or decreases by a constant number, known as the common difference. To find the sum of the sequence, we use a specific formula that helps add up all the terms without having to list or manually add each one. This formula is:\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]Let's break this down:

• \(S_n\) is the sum of the sequence.
• \(n\) represents the number of terms.
• \(a_1\) is the first term of the sequence.
• \(a_n\) is the last term of the sequence.

For the log pile problem, we know:

• First term \(a_1 = 25\)
• Last term \(a_n = 1\)
• Number of terms \(n = 25\)

Substituting these into our formula gives us:\[ S_{25} = \frac{25}{2} \cdot (25 + 1) = 25 \cdot 13 = 325 \]This result tells us there are 325 logs in total. Using the formula simplifies and verifies our calculation.

Number of Terms in Sequence

Determining the number of terms in an arithmetic sequence is crucial for calculating the total sequence sum or working with any related formula. To find the number of terms \(n\), you can use the following formula which involves the common difference \(d\) and the last term \(a_n\):\[ a_n = a_1 + (n-1) \cdot d \]Where:

• \(a_1\) is the first term.
• \(d\) is the common difference.
• \(a_n\) is the known last term.

For the exercise at hand:

• First term \(a_1 = 25\)
• Common difference \(d = -1\)
• Last term \(a_n = 1\)

Plug these into the formula:\[ 1 = 25 + (n-1)(-1) \]Simplify to find \(n = 25\). This tells us there are 25 layers of logs in the pile.

Common Difference in Sequence

The common difference in an arithmetic sequence represents how much each term increases or decreases relative to the previous term. Understanding this concept helps to predict future terms or backtrack to find past terms.In formula terms, the common difference \(d\) is expressed as:\[ d = a_{n} - a_{n-1} \]Where \(a_{n}\) is the current term and \(a_{n-1}\) is the previous term. In our logs example, we observe:

• The first term \(a_1 = 25\)
• The second term is 24, which is built upon by subtracting 1 from 25

Thus, \(d = 24 - 25 = -1\). The negative sign indicates a decrease. Repeated subtraction by 1 stacks up to form the sequence from 25 logs down to 1 log.Knowing the common difference lets you easily fill in missing terms or understand how the sequence will extend or diminish.