Such numbers form an arithmetic sequence: $3, 10, 17, \dots$, with first term $a = 3$, common difference $d = 7$.
["Understanding Arithmetic Sequences: An Example Using $3, 10, 17, \dots$", "An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. This concept is fundamental in mathematics and appears frequently in patterns, finance, computer science, and everyday problem-solving.", "One classic example of an arithmetic sequence is $3, 10, 17, \dots$. In this sequence, the first term $a$ is $3$, and the common difference $d$ is $7$. Understanding how such sequences work helps in recognizing patterns, predicting future values, and solving real-world problems efficiently.", "### What Makes This a Valid Arithmetic Sequence?", "By definition, an arithmetic sequence follows the rule:\n$$\na_n = a + (n - 1)d\n$$\nwhere:\n- $a_n$ is the $n^\ ext{th}$ term,\n- $a$ is the first term ($3$ in this case),\n- $d$ is the common difference ($7$),\n- $n$ is the position of the term.", "Let’s verify the first few terms:\n- $a_1 = 3 + (1 - 1)\cdot 7 = 3$\n- $a_2 = 3 + (2 - 1)\cdot 7 = 3 + 7 = 10$\n- $a_3 = 3 + (3 - 1)\cdot 7 = 3 + 14 = 17$", "This confirms that the sequence $3, 10, 17, \dots$ is indeed an arithmetic sequence with $a = 3$ and $d = 7$.", "### Formula for the $n^\ ext{th}$ Term", "Using the general formula:\n$$\na_n = 3 + (n - 1)\cdot 7 = 7n - 4\n$$\nSo, the $n^\ ext{th}$ term of the sequence is $a_n = 7n - 4$. This allows anyone to quickly compute any term without listing all previous values.", "### Finding Any Term Without Calculating All Previous Ones", "Suppose you want the 15th term. Instead of computing $a_{14} + 7$, simply substitute $n = 15$:\n$$\na_{15} = 7 \cdot 15 - 4 = 105 - 4 = 101\n$$\nThis efficiency is a powerful advantage of arithmetic sequences.", "### Real-World Applications", "Arithmetic sequences model situations where progress occurs at a constant rate. Examples include:\n- Saving a fixed amount of money each month\n- Calculating cumulative growth with constant increments\n- Organizing items spaced regularly in time or space\n- Programming loops with stepwise progression", "---", "### Summary", "The sequence $3, 10, 17, \dots$ exemplifies a classic arithmetic progression with first term $a = 3$ and common difference $d = 7$. By applying the formula for the $n^\ ext{th}$ term, $a_n = 7n - 4$, we can efficiently analyze and predict any term in the sequence. Understanding such patterns aids in mathematical reasoning and practical problem-solving across diverse fields.", "Whether you're a student learning foundational math or a professional applying sequences in data analysis, recognizing arithmetic patterns unlocks powerful tools for analysis and planning."]
