Question**: A sequence is defined by \( a_n = 3n + 2 \). What is the sum of the first 10 terms of this sequence? - NBX Soluciones
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:
Image Gallery
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 \( a_n = 3n + 2 \):
🔗 Related Articles You Might Like:
📰 Recommended Computer Games 📰 Pecker Game 📰 Grand Theft Auto 5 Pc Game Download 📰 Trevor St John 4601170 📰 Eczema On Eyelids Cure 6219274 📰 Download The Disney Appyour Familys Ultimate Binge Watching Secret 8568669 📰 Free Comic Book Day 2025 2271156 📰 Wells Fargo Bank Waldorf Md 9119587 📰 Perfume And Perfume 7220021 📰 Marvel Strong Characters 2768978 📰 Why This Mac Bid Strategy Is Sweeping The Tech Communityyouve Got To See It 6876238 📰 You Wont Believe How Easy Mundo Build Rewrote The Rules Of Gaming Construction 2335607 📰 Transform Your Spreadsheets Color Every Second Row At Once In Excel 7870755 📰 Treated Lumber Raised Garden 4231642 📰 Littoral Zone 3356947 📰 Verkehr 663691 📰 You Wont Believe How Aprs Fi Changed Live Streaming Forever 1481271 📰 Can U Do Split Screen On Fortnite 2169761Final Thoughts
- 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:
\[
a_{10} = 3(10) + 2 = 30 + 2 = 32
\]
