Question**: A sequence is defined as $ a_n = 3n + 2 $. What is the 10th term of the sequence? - RoadRUNNER Motorcycle Touring & Travel Magazine

Understanding the Sequence $ a_n = 3n + 2 $: What Is the 10th Term?

In mathematics, sequences define ordered lists of numbers following a specific rule. One common type is an arithmetic sequence, where each term increases by a fixed amount. The sequence defined by $ a_n = 3n + 2 $ is a classic example of a linear sequence with a constant difference.

What is the 10th Term of the Sequence?

Understanding the Context

To find the 10th term ($ a_{10} $) in the sequence given by the formula $ a_n = 3n + 2 $, simply substitute $ n = 10 $:

\[
a_{10} = 3(10) + 2 = 30 + 2 = 32
\]

So, the 10th term is 32.

How Does This Sequence Work?

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Key Insights

The general term $ a_n = 3n + 2 $ shows a clear linear relationship. For each positive integer $ n $, multiplying $ n $ by 3 and adding 2 generates the corresponding term:

When $ n = 1 $: $ a_1 = 3(1) + 2 = 5 $
- When $ n = 2 $: $ a_2 = 3(2) + 2 = 8 $
- When $ n = 3 $: $ a_3 = 3(3) + 2 = 11 $
- …
- When $ n = 10 $: $ a_{10} = 32 $

This consistency allows us to confidently compute any term directly using the formula.

Why Is This Sequence Important?

Understanding how to calculate terms in sequences like $ a_n = 3n + 2 $ is essential in algebra, computer science, and real-world applications. Such formulas help model linear growth, plan budgets, or simulate patterns in data.

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Final Thoughts

In summary:
The 10th term of the sequence defined by $ a_n = 3n + 2 $ is 32. By plugging $ n = 10 $ into the formula, any term in this arithmetic sequence can be found quickly and accurately.

If you're exploring sequences, remember: linear formulas like $ a_n = an + b $ offer an easy pathway to compute individual terms and recognize patterns across the sequence.