Arithmetic Progressions and Multiples of 1,500

Multipliers and Linear Sequences

An arithmetic sequence is formed by multiplying consecutive integer values by a fixed constant coefficient. In this linear scaling model, the base unit step size is $1,500$, representing a constant rate of change. As the independent input variable $x$ increments by $1$, the resulting output $y$ increases uniformly by $1,500$. This relationship is governed by the linear equation $y = 1,500 \times x$, where $x$ spans integers from $2$ through $24$.

The sequence illustrates strict proportional scaling across a range of values. Initial observation begins at the integer value $2$, which yields $3,000$, and progresses sequentially to $24$, which produces $36,000$. Such constant incrementation demonstrates the core characteristics of an arithmetic sequence, wherein the common difference $d$ equals $1,500$.

Tabular Breakdown of Sequential Multiples

For the initial integer input $x = 2$, multiplying by the constant $1,500$ results in an output value of $3,000$. Incrementation to $x = 3$ yields $4,500$. Advancing to $x = 4$ yields $6,000$.

Continuing the progression, an input of $x = 5$ produces $7,500$. At $x = 6$, the total reaches $9,000$. For an input of $x = 7$, the resulting value is $10,500$. At $x = 8$, the calculation yields $12,000$. When $x = 9$, the calculated total is $13,500$. At the benchmark value $x = 10$, the output is $15,000$.

Progressing into the higher integer range, an input of $x = 11$ gives $16,500$. For $x = 12$, the value reaches $18,000$. At $x = 13$, the output increases to $19,500$. Setting $x = 14$ yields $21,000$. An input of $x = 15$ produces $22,500$.

In the upper tier of the sequence, an input of $x = 16$ generates $24,000$. With $x = 17$, the result is $25,500$. For $x = 18$, the calculated value is $27,000$. Moving to $x = 19$, the total reaches $28,500$. At $x = 20$, the output hits $30,000$.

For the final set of calculations, an input of $x = 21$ yields $31,500$. With $x = 22$, the total is $33,000$. At $x = 23$, the product is $34,500$. Finally, at $x = 24$, the maximum term evaluated in the sequence equals $36,000$.

Mathematical Formulation and Sequence Summation

The general term of an arithmetic sequence is defined by the standard explicit formula $a_n = a_1 + (n - 1) \times d$, where $a_1$ is the first term of the sequence, $n$ is the term position, and $d$ is the common difference. Substituting $a_1 = 1,500$ and $d = 1,500$ yields the simplified model $a_n = 1,500 \times n$. For the sequence running from index $n = 2$ to index $n = 24$, the total number of terms $N$ is given by $N = 24 - 2 + 1 = 23$.

The summation of all $23$ terms in this specific range can be determined using the arithmetic series sum formula $S_N = \frac{N}{2} \times (a_{\text{start}} + a_{\text{end}})$. Inserting the start term $a_2 = 3,000$ and end term $a_{24} = 36,000$ into the formula gives $S_{23} = \frac{23}{2} \times (3,000 + 36,000)$. Simplifying the expression leads to $S_{23} = 11.5 \times 39,000 = 448,500$.

This series demonstrates both linear growth and constant step size, serving as a fundamental foundation for proportional reasoning, tabular interpolation, and step-function modeling across physical and mathematical contexts.