Solution: Let the number of shelves be \( n \). The number of panels forms an arithmetic sequence: 7, 11, 15, ..., with first term \( a = 7 \) and common difference \( d = 4 \). The sum of the first \( n \) terms is: - Coaching Toolbox
Solution: Let the Number of Shelves Be \( n \)—Understanding the Panel Arrangement in an Arithmetic Sequence
Solution: Let the Number of Shelves Be \( n \)—Understanding the Panel Arrangement in an Arithmetic Sequence
In modular design and retail shelving, optimizing space while maintaining aesthetic harmony is crucial. One practical solution involves arranging panels in a structured arithmetic sequence—a method ideal for uniform display and scalable construction. This article explores a key mathematical solution for shelves arranged using a specific arithmetic progression, helping designers and engineers calculate total panel count efficiently.
Let the number of shelves be \( n \). The width or placement of panels follows an arithmetic sequence starting with first term \( a = 7 \) and common difference \( d = 4 \). This means the number of panels at each shelf increases consistently: 7, 11, 15, ..., forming a clear pattern.
Understanding the Context
The Arithmetic Sequence in Shelf Design
The general formula for the \( k \)-th term of an arithmetic sequence is:
\[
a_k = a + (k - 1)d
\]
Substituting \( a = 7 \) and \( d = 4 \):
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Key Insights
\[
a_k = 7 + (k - 1) \cdot 4 = 4k + 3
\]
The total number of panels across \( n \) shelves is the sum of the first \( n \) terms of this sequence, denoted \( S_n \). The sum of the first \( n \) terms of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} (a + a_n)
\]
where \( a_n \) is the \( n \)-th term:
\[
a_n = 4n + 3
\]
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Substituting into the sum formula:
\[
S_n = \frac{n}{2} \left( 7 + (4n + 3) \right) = \frac{n}{2} (4n + 10)
\]
Simplify the expression:
\[
S_n = \frac{n}{2} \cdot 2(2n + 5) = n(2n + 5)
\]
Thus, the number of panels required is:
\[
\boxed{S_n = n(2n + 5)}
\]
Practical Implications
This formula allows precise planning: for \( n \) shelves, total panel count is \( n \ imes (2n + 5) \), combining simplicity with scalability. For instance:
- With \( n = 1 \): \( S_1 = 1 \ imes (2 \cdot 1 + 5) = 7 \) panels ✔
- With \( n = 5 \): \( S_5 = 5 \ imes (2 \cdot 5 + 5) = 5 \ imes 15 = 75 \) panels ✔
- With \( n = 10 \): \( S_{10} = 10 \ imes 25 = 250 \) panels ✔
Using arithmetic progression ensures visual alignment, timely construction, and efficient inventory management in retail or display applications.
