Substitute \(n = 10\) into the formula: \(a_{10} = 2(10) + 3 = 20 + 3 = 23\).
["Understanding the Substituted Formula: Evaluating (a_{10} = 2(10) + 3)", "When studying sequences, mathematical formulas often play a central role in determining specific terms. One such expression is the formula (a_n = 2n + 3), commonly used to generate terms in a linear sequence. A frequently asked question is what happens when substituting (n = 10) into this formula — specifically, why it results in (a_{10} = 23). This article explores the substitution process, explains the formula clearly, and shows how it directly leads to the correct result.", "### The General Formula: (a_n = 2n + 3)", "In sequences defined by linear formulas, (a_n) represents the (n)-th term based on a simple linear relationship. Here, (a_n = 2n + 3) means:", "- Each term is generated by multiplying the term position (n) by 2, then adding 3.\n- For (n = 1), (a_1 = 2(1) + 3 = 5)\n- For (n = 2), (a_2 = 2(2) + 3 = 7), and so on.", "This pattern continues indefinitely, defining an arithmetic sequence with a constant difference of 2 between consecutive terms.", "### Substituting (n = 10)", "To find the 10th term, we substitute (n = 10) directly into the formula:", "[\na_{10} = 2(10) + 3\n]", "Carrying out the multiplication first:", "[\na_{10} = 20 + 3\n]", "Then applying the addition:", "[\na_{10} = 23\n]", "### Why This Substitution Works", "Substituting a specific value for (n) reduces the general formula to an arithmetic calculation. The term (2 \ imes 10 = 20) scales the position in the sequence, and adding 3 shifts the result, reflecting both the linear growth and the constant addition inherent to the formula. This substitution is straightforward yet powerful, enabling precise term calculation without memorizing the entire sequence.", "### Applications and Real-World Relevance", "Formulas like (a_n = 2n + 3) appear across various fields:", "- Education: Modeling progress in skill acquisition or lexical growth\n- Finance: Calculating fixed-rate growth or payments\n- Science and Engineering: Representing linear trends in data or sensor outputs", "Understanding the mechanics of substitution helps users verify results, adapt formulas, and apply them beyond simple sequences.", "### Summary", "Evaluating (a_{10} = 2(10) + 3) is a clear example of applying a linear formula to find a specific term. The computation replaces (n) with 10, performs basic arithmetic, and yields:", "[\na_{10} = 23\n]", "This formula and substitution method offer a reliable, scalable way to navigate many mathematical and real-world problems involving linear progressions.", "---", "Keywords: substitute (n = 10), (a_{10} = 2(10) + 3), linear formula, arithmetic sequence, term calculation, math education, formula substitution."]
