Check: $n_{14} = 7(14) - 4 = 98 - 4 = 94 \leq 100$, and $n_{15} = 7(15) - 4 = 105 - 4 = 101 > 100$.
["Understanding the Mathematical Check: $n_{14} = 7(14) - 4 = 94$ and $n_{15} = 7(15) - 4 = 101 > 100$", "In number theory and pattern-based problem solving, checking sequences defined by simple linear formulas is a common technique to identify when values fall within a defined range. A recent example involves evaluating two terms in a sequence defined as $n_k = 7k - 4$, focusing on whether $n_{14}$ and $n_{15}$ are within the threshold of 100.", "### The Formula and Computation", "Let’s start with $n_{14}$:", "$$\nn_{14} = 7 \ imes 14 - 4 = 98 - 4 = 94\n$$", "Now compute $n_{15}$:", "$$\nn_{15} = 7 \ imes 15 - 4 = 105 - 4 = 101\n$$", "Both expressions follow the linear pattern $n_k = 7k - 4$, making direct substitution an efficient method.", "### Evaluating the Condition", "We compare both results against the upper limit of 100:", "- $n_{14} = 94 \leq 100$ – True\n- $n_{15} = 101 > 100$ – False", "This check serves as a clear decision point: $n_{14}$ lies within the defined bound, while $n_{15}$ exceeds it.", "### Application in Problem Solving", "Such evaluations help in classification, selection, and boundary setting in both educational problems and algorithmic logic. They demonstrate how modular patterns and arithmetic sequences simplify checking limits without complex computations.", "### Conclusion", "The check confirms that $n_{14} = 94$ satisfies the inequality $n_k \leq 100$, but $n_{15} = 101$ does not. This simple number verification illustrates a foundational approach in pattern recognition and mathematical reasoning—essential skills in fields ranging from programming to competitive mathematics.", "Key Takeaway: Using linear formulas like $7k - 4$, boundary checks with concise arithmetic (e.g., $94 \leq 100$, $101 > 100$) offer clear, efficient validation within numerical ranges.", "---", "Keywords: $n_k = 7k - 4$, mathematical check, $n_{14} = 94$, $n_{15} = 101$, inequality evaluation, linear patterns, number theory example, threshold comparison.\nMeta description: Verify the values $n_{14} = 7(14) - 4$ and $n_{15} = 7(15) - 4$ using direct computation and boundary testing to determine which lie within $100$."]
