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:

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 $:

Arithmetic Sequence Formula - Learn the Formulas for Arithmetic Sequence

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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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Final Thoughts

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.