Question**: A sequence is defined as \( a_n = 3n + 2 \). What is the 10th term of the sequence? - Abu Waleed Tea
Understanding the Sequence \( a_n = 3n + 2 \): What Is the 10th Term?
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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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.
