Question**: A sequence is defined by \( a_n = 3n + 2 \). What is the sum of the first 10 terms of this sequence? - Decision Point
Sum of the First 10 Terms of the Sequence Defined by \( a_n = 3n + 2 \)
Sum of the First 10 Terms of the Sequence Defined by \( a_n = 3n + 2 \)
Understanding arithmetic sequences is fundamental in mathematics, especially when calculating cumulative sums efficiently. One such sequence is defined by the formula \( a_n = 3n + 2 \), where every term increases consistently. In this article, we explore how to find the sum of the first 10 terms of this sequence using a step-by-step approach grounded in mathematical principles.
Understanding the Context
Understanding the Sequence
The sequence is defined by the closed-form expression:
\[
a_n = 3n + 2
\]
This linear expression describes an arithmetic sequence, where each term increases by a constant difference. Let’s compute the first few terms to observe the pattern:
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Key Insights
- \( a_1 = 3(1) + 2 = 5 \)
- \( a_2 = 3(2) + 2 = 8 \)
- \( a_3 = 3(3) + 2 = 11 \)
- \( a_4 = 3(4) + 2 = 14 \)
- ...
From this, we see that the sequence begins: 5, 8, 11, 14, ..., increasing by 3 each time.
Identifying the First Term and Common Difference
From \( a_n = 3n + 2 \):
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- First term (\( a_1 \)):
\[
a_1 = 3(1) + 2 = 5
\]
- Common difference (\( d \)): The coefficient of \( n \) — here \( d = 3 \).
Since this is an arithmetic sequence, the sum of the first \( n \) terms is given by the formula:
\[
S_n = \frac{n}{2}(a_1 + a_n)
\]
where \( a_n \) is the \( n \)-th term.
Step 1: Compute the 10th Term (\( a_{10} \))
Using the formula:
\[
a_{10} = 3(10) + 2 = 30 + 2 = 32
\]
