Question**: A sequence is defined by \( a_n = 3n + 2 \). What is the sum of the first 10 terms of this sequence? - RoadRUNNER Motorcycle Touring & Travel Magazine
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:
📰 Grand Rising Exposed: Why This Trend is Dominating Every Industry Who’s Watch! 📰 Grand Rising: The Rise That Everyone Talks About—Discover What’s Fueling It Now! 📰 Grand Rising Explained: The Shocking Reasons Behind Its Explosive Popularity! 📰 Why Arkvx Is Riding The Tech Waveits Stock Is About To Surge Heres How 5074221 📰 Unlock Your Inner Musician The Ultimate Music Note App That No One Talks About 8248190 📰 Medtronics Stock Unveils Explosive 200 Surgeheres What It Means For Investors 9928353 📰 The Ultimate Pdp Controller Hub Revealedupgrade Your Tech Now 8738503 📰 7000 9324071 📰 How This Tiny Wordgrid Can Boost Your Vocabulary Overnight 2784665 📰 Mac League Of Legends 📰 This Satin Dress Is So Beautiful Fashion Influencers Are Overhead Hiring It For Their Next Glow Up 6544377 📰 Sudden Decision Spiderman 2 Suits And People Are Shocked 📰 Substack Alternatives 📰 Fidelity Asset Allocation The Smart Investors Formula You Cant Afford To Miss 7321569 📰 Roblox Dandys World 253664 📰 Steam Half Sword 📰 Is Your Payment Frozen Mypaymentsplus Cracks The Mystery In Seconds 2156389 📰 Data Shows Stuck Together And The Reaction SpreadsFinal 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:
\[
a_{10} = 3(10) + 2 = 30 + 2 = 32
\]
