Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Log pile A pile of logs has 24 logs in the bottom layer, 23 in the second layer, 22 in the third, and so on. The top layer contains 10 logs. Find the total number of logs in the pile.

Step-by-Step Solution

Answer

The total number of logs is 255.

Step 1: Identify the Sequence

The number of logs decreases by one in each successive layer. The sequence of logs can be identified as: 24, 23, 22, ... , 10. This is an arithmetic sequence.

Step 2: Determine the Sequence Parameters

For an arithmetic sequence, the common difference, \( d \), is -1 (since each layer has one log less than the previous one). The first term, \( a_1 \), is 24.

Step 3: Determine the Number of Terms

The last term of the sequence, \( a_n \), is 10. Using the formula for the nth term of an arithmetic sequence \( a_n = a_1 + (n-1) \cdot d \), we find the number of terms, \( n \): \( 10 = 24 + (n-1)(-1) \). Solving for \( n \) gives \( n = 15 \).

Step 4: Calculate the Total Number of Logs

The sum of an arithmetic sequence can be calculated using the formula \( S_n = \frac{n}{2} (a_1 + a_n) \). Substituting the values, \( S_{15} = \frac{15}{2} (24 + 10) \), gives a sum of \( S_{15} = \frac{15}{2} \times 34 = 255 \).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is what drives the pattern from one number to the next. It's the consistent amount that is added (or subtracted) as you move through the sequence. This concept is quite simple yet powerful.
The common difference, denoted as \( d \), is calculated by subtracting any term from the one before it.

• If your sequence is increasing, \( d \) will be positive.
• If it's decreasing, as in our log pile example, \( d \) will be negative.

In our exercise, each layer of logs contains one fewer log than the layer below it. So, the common difference is \( -1 \). This means for each step up in layers, we lose one log. Understanding \( d \) helps us determine other aspects of the sequence, like the nth term.

Arithmetic Series

An arithmetic series involves the sum of an arithmetic sequence. It's like adding up the terms in your sequence.
Imagine you have a staircase with steps made from the numbers of your sequence—you want to find the height by adding all these steps together. In our log pile problem, the sequence is: 24, 23, 22, ..., 10.

To find the series sum, you add all these numbers: 24, 23, 22, etc., until you reach 10. Calculating series manually is manageable for short sequences but can be lengthy otherwise. Thankfully, there's a formula to make life easier! The arithmetic series formula allows us to efficiently compute the sum without listing and adding each term individually.

Nth Term Formula

Using the nth term formula of an arithmetic sequence is key to unlocking its hidden patterns.
This formula helps you locate a specific term in the sequence without listing every preceding number.

• The formula is: \( a_n = a_1 + (n-1) \cdot d \).
• \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

In our log pile context, the first term \( a_1 \) is 24 logs, the common difference \( d \) is -1, and the last term \( a_n \) is 10 logs. We plug these into the formula to find that there are 15 layers. Understanding how to manipulate and solve this formula is essential for working efficiently with sequences.

Sequence Sum Formula

The sequence sum formula is your best friend when it comes to finding the total of an arithmetic series without tedious counting. It lets you sum all the sequence's terms in just a few quick steps.

• The formula is: \( S_n = \frac{n}{2} (a_1 + a_n) \).
• \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the last term.

For the log pile scenario, we learned there are 15 layers, with 24 logs at the base and 10 at the top. Plugging these into the formula, we calculate: \( S_{15} = \frac{15}{2} \times (24 + 10) = 255 \). This magical formula quickly gives us the sum without needing to add each log count layer by layer.