10.3
Arithmetic Sequences
An arithmetic sequence is a list of numbers where each term increases or decreases by the same fixed number, known as the common difference. Consider a pile of poles. The first layer contains 25 poles, and the number of poles continues to decrease by 1 in each successive layer.
Given that the pile has 12 layers, the goal is to find the total number of poles.
This arrangement forms an arithmetic sequence, as the number of poles decreases by a constant amount from one layer to the next.
In this scenario, the number of poles in the 12th layer is calculated using the formula for the nth term of an arithmetic sequence, based on the first term, the common difference, and the number of layers. The values of these terms are then substituted into the formula, which simplifies to 25 minus 11, giving 14 poles in the 12th layer.
The total number of poles in the pile, known as the partial sum of the sequence, is then calculated by taking the average of the number of poles in the first and last layers and multiplying it by the total number of layers. It is termed a partial sum since only the first 12 terms of the sequence are added. This results in a partial sum of 12 multiplied by the average of 25 and 14, yielding 234 poles.
An arithmetic sequence is a structured arrangement of numbers where each term is derived by adding a constant value, known as the common difference, to the previous term. This consistent pattern allows for the efficient computation of any term within the sequence as well as the cumulative sum of multiple terms. The formula for finding the nth term of an arithmetic sequence is:
Here, aₙ represents the nth term of the sequence, a is the first term, d is the common difference, and n is the term number or position in the sequence. This equation is essential for determining the value of any term without listing all preceding terms. To compute the total of the first n terms, known as the partial sum, either of the following formulas is used:
In these expressions, Sₙ denotes the sum of the first n terms, and aₙ again refers to the nth term, derived using the earlier formula. These formulas offer a concise and systematic approach for analyzing regularly spaced numerical patterns in both theoretical and practical applications.
An arithmetic sequence is a list of numbers where each term increases or decreases by the same fixed number, known as the common difference. Consider a pile of poles. The first layer contains 25 poles, and the number of poles continues to decrease by 1 in each successive layer.
Given that the pile has 12 layers, the goal is to find the total number of poles.
This arrangement forms an arithmetic sequence, as the number of poles decreases by a constant amount from one layer to the next.
In this scenario, the number of poles in the 12th layer is calculated using the formula for the nth term of an arithmetic sequence, based on the first term, the common difference, and the number of layers. The values of these terms are then substituted into the formula, which simplifies to 25 minus 11, giving 14 poles in the 12th layer.
The total number of poles in the pile, known as the partial sum of the sequence, is then calculated by taking the average of the number of poles in the first and last layers and multiplying it by the total number of layers. It is termed a partial sum since only the first 12 terms of the sequence are added. This results in a partial sum of 12 multiplied by the average of 25 and 14, yielding 234 poles.
From Chapter 10:
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