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Chapter 10: Problem 62

If 25 logs are laid side by side on the ground, and 24 logs are placed on top of those, and 23 logs are placed on the 3 rd row, and the pattern continues until there is a single log on the 25th row, how many logs are in the stack?

Short Answer

Expert verified
There are 325 logs in the stack.

Step by step solution

01

Understand the Problem

The problem describes a pyramid-like structure made of logs. The first row contains 25 logs, the second row has 24 logs, the third has 23 logs, and this sequence continues until the 25th row, which has only 1 log.
02

Identify the Sequence

We recognize that this is an arithmetic sequence where the number on each row decreases by one. The sequence starts at 25 and ends at 1.
03

Use the Formula for Sum of an Arithmetic Series

The formula for the sum of an arithmetic series is \( S = \frac{n}{2} (a + l) \) where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.
04

Substitute Known Values into the Formula

Identify the parameters: \(n = 25\), \(a = 25\), and \(l = 1\). Substituting these into the formula, we get:\[ S = \frac{25}{2} (25 + 1) \]
05

Calculate the Sum

First, calculate the value inside the parentheses: \(25 + 1 = 26\).Then, calculate \(\frac{25}{2} \times 26 = 12.5 \times 26 = 325\).Thus, the total number of logs is 325.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Arithmetic Sequences
An arithmetic sequence is a sequence of numbers where each successive term is derived by adding or subtracting a constant value, known as the common difference, to the previous term. In the exercise, logs are arranged such that each row contains one less log than the previous row.
For instance:
  • The first row has 25 logs.
  • The second row has 24 logs.
  • The third row has 23 logs.
This continues until the last row, which has a single log. The common difference in this sequence is -1. This decrease forms a predictable pattern, making it an arithmetic sequence. Recognizing these sequences is crucial, especially in precalculus problems, as it allows us to apply specific formulas to find sums or predict future terms.
Calculating the Sum of an Arithmetic Series
The sum of an arithmetic series involves adding together all terms in the sequence. When we have identified it as an arithmetic sequence, we can use the formula for finding the sum, \[ S = \frac{n}{2} (a + l) \]where:
  • \( S \) is the sum of the series.
  • \( n \) is the number of terms.
  • \( a \) is the first term.
  • \( l \) is the last term.
This formula provides a compact way to calculate the sum without manually adding each term, which can be tedious with larger sequences. Applying this formula to our exercise ensures a quick and easy computation of the total number of logs.
Recognizing Mathematical Patterns
In solving mathematical problems, recognizing patterns simplifies the approach. In the given problem, recognizing the decreasing number of logs as a mathematical pattern 鈥� specifically an arithmetic sequence 鈥� allows for the use of established mathematical techniques and formulas.
Patterns exist in many forms both familiar and complex:
  • They can be visual, like geometric shapes.
  • They can be numerical, like sequences and series.
  • Or a mix of both, occurring in real-world problems and abstract math.
Identifying patterns helps streamline problem-solving as recognized patterns provide a pathway to solutions.
Solving Precalculus Problems with Sequences
Precalculus often involves problems related to sequences and series, like the log stacking puzzle. These problems test your ability to identify the type of sequence and apply the correct formulas.
Some tips for solving such problems include:
  • Thoroughly read through the problem to identify the pattern or sequence described.
  • Determine the important values: starting number, the pattern or rule of progression, and a stopping point.
  • Apply appropriate formulas, like the sum of an arithmetic series, which simplifies the calculation process.
By approaching precalculus problems with these structured steps, students can tackle exercises more effectively and build a stronger foundation for calculus and beyond.

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Most popular questions from this chapter

Upon graduation Jessica receives a commission from the U.S. Navy to become an officer and a 20,000 dollars signing bonus for selecting aviation. She puts the entire bonus in an account that earns \(6 \%\) interest compounded monthly. The balance in the account after \(n\) months is $$A_{n}=20,000\left(1+\frac{0.06}{12}\right)^{n} \quad n=1,2,3, \ldots$$ Her commitment to the Navy is 6 years. Calculate \(A_{72} .\) What does \(A_{72}\) represent?

In calculus, we study the convergence of sequences. A sequence is convergent when its terms approach a limiting value. For example, \(a_{n}=\frac{1}{n}\) is convergent because its terms approach zero. If the terms of a sequence satisfy \(a_{1} \leq a_{2} \leq a_{3} \leq \ldots \leq a_{n} \leq \ldots\) the sequence is monotonic nondecreasing. If \(a_{1} \geq a_{2} \geq a_{3} \geq \ldots \geq a_{n} \geq \ldots,\) the sequence is monotonic nonincreasing. Classify each sequence as monotonic or not monotonic. If the sequence is monotonic, determine whether it is nondecreasing or nonincreasing. $$a_{n}=\sin \left(\frac{n \pi}{4}\right)$$

Determine whether each statement is true or false. $$\sum_{k=0}^{n} c x^{k}=c \sum_{k=0}^{n} x^{k}$$

Upon graduation Sheldon decides to go to work for a local police department. His starting salary is 30,000 dollars per year, and he expects to get a \(3 \%\) raise per year. Write the recursion formula for a sequence that represents his annual salary after \(n\) years on the job. Assume \(n=0\) represents his first year making 30,000 dollars.

Simplify each ratio of factorials. $$\frac{(n+2) !}{n !}$$
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