Question**: A sequence is defined by \( a_n = 3n + 2 \). What is the sum of the first 10 terms of this sequence?
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 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:
- \( a_1 = 3(1) + 2 = 5 \)- \( a_2 = 3(2) + 2 = 8 \)- \( a_3 = 3(3) + 2 = 11 \)- \( a_4 = 3(4) + 2 = 14 \)- ...
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 \( a_n = 3n + 2 \):
- 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:
\[a_{10} = 3(10) + 2 = 30 + 2 = 32\]
Step 2: Calculate the Sum of the First 10 Terms (\( S_{10} \))
Using the sum formula with \( n = 10 \), \( a_1 = 5 \), and \( a_{10} = 32 \):
\[S_{10} = \frac{10}{2}(5 + 32) = 5 \ imes 37 = 185\]
Conclusion
The sum of the first 10 terms of the sequence defined by \( a_n = 3n + 2 \) is 185. This method—leveraging the formula for the \( n \)-th term and the arithmetic sum—provides a fast and efficient way to compute totals without adding each term individually.
Whether you're solving math problems, writing code, or analyzing patterns, understanding how to sum such sequences is a valuable skill. Use this approach as a template for any linear sequence: identify \( a_1 \), compute \( a_{10} \), and apply the sum formula.
Keywords: sequence sum, arithmetic series formula, \( a_n = 3n + 2 \), sum of first 10 terms, \( S_{10} \), mathematical sequence, lesson on summing sequences, linear recursive sequence.Meta Description: Learn how to calculate the sum of the first 10 terms of the arithmetic sequence defined by \( a_n = 3n + 2 \). Step-by-step guide with sum formula and verification.
