10.3
Arithmetic Sequences
An arithmetic sequence is a structured arrangement of numbers where each term is derived by adding a constant value, known as the common difference, t…
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.
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Q1: What is a common difference in an arithmetic sequence?
The common difference is the fixed constant value added to or subtracted from each term to get the next term in an arithmetic sequence. In the pole pile example, the common difference is -1 because the number of poles decreases by one in each successive layer. This consistent pattern allows you to predict any term in the sequence without listing all preceding terms.
Q2: How do you find the nth term of an arithmetic sequence?
Use the formula aₙ = a + (n - 1)d, where aₙ is the nth term, a is the first term, d is the common difference, and n is the term position. For the pole pile with 25 poles in layer 1 and common difference -1, the 12th layer contains 25 + (12 - 1)(-1) = 14 poles. This formula eliminates the need to calculate every preceding term.
Q3: What is the partial sum formula for an arithmetic sequence?
The partial sum Sₙ equals n times the average of the first and last terms: Sₙ = n(a + aₙ)/2. For the 12-layer pole pile, the total is 12 × (25 + 14)/2 = 234 poles. This formula efficiently computes the sum of the first n terms without adding each term individually, making it practical for large sequences.
Q4: Why is the pole pile arrangement considered an arithmetic sequence?
The pole pile forms an arithmetic sequence because each layer contains a constant decrease of one pole from the previous layer. This consistent pattern of change—the common difference—is the defining characteristic of arithmetic sequences. The structured arrangement allows systematic calculation of any layer's pole count and the total across all layers.
Q5: How does the partial sum differ from the total sum of an infinite sequence?
A partial sum calculates only the first n terms of a sequence, as shown when finding the total of 12 layers in the pole pile. The partial sum formula Sₙ = n(a + aₙ)/2 applies to a finite number of terms. This approach is practical for real-world applications where you need the cumulative total of a specific portion rather than an entire infinite sequence.
Q6: Can you use arithmetic sequences to solve real-world stacking problems?
Yes, arithmetic sequences model real-world stacking scenarios like the pole pile, where items decrease or increase by a constant amount per layer. By identifying the first term, common difference, and number of layers, you can calculate individual layer quantities using the nth term formula and total inventory using summation notation. This systematic approach simplifies inventory and structural calculations.
Q7: What is the relationship between arithmetic sequences and geometric sequences?
Arithmetic sequences use addition of a constant common difference, while geometric sequences use multiplication by a constant ratio. Both are structured numerical patterns that allow efficient term and sum calculations. Understanding arithmetic sequences provides foundational knowledge for comparing with geometric sequences and recognizing which pattern applies to different real-world or theoretical problems.