Question**: A sequence is defined by \( a_n = 3n + 2 \). What is the sum of the first 10 terms of this sequence? - Dachbleche24
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:
📰 From Box to Baked Perfection: Corn Pudding with Jiffy Mix That Wows! 📰 This Hidden Corner Cabinet Secret Will Transform Your Kitchen Storage! 📰 You Won’t Believe How Much Space This Corner Bar Cabinet Saves—Used It and Regretted NOT Buying One! 📰 You Best Not Try These Treatstheir Secret Ingredient Will Shock You 📰 You Can Never Ignore This Powerful Secret Behind Ple It Pays Off In Mysterious Ways 📰 You Could Have Won Big But Now Everything Has Turned Into A Risk 📰 You Deserve To Know Dry Sockets Warning Signs Vs Soft Recoverys Gentle Reality 📰 You Desperate For Impatiencebut True Patience Reveals Hidden Truths 📰 You Desperate For Pork Belly Heres Your Chanceits Closer Than You Think 📰 You Desperately Need These Hidden Pink Ugg Slippers Under Your Shoes 📰 You Didnt Just Get A Smileyou Unlocked Perfection 📰 You Didnt Just Get Pierced Now Every Spin Of The Ring Sends Waves Through Your Soul 📰 You Didnt Know Daycare Could Be This Easypart Time Setup Yes Or No 📰 You Didnt Know How To Secure Your Playmetrics Loginnow Fix It Fast 📰 You Didnt Know Perfumeto Worked Like Magicheres How 📰 You Didnt Know Popcorn Ceilings Were Killing But This Tool Exposes Every Trace 📰 You Didnt Know Pressure Socks Could Fix Your Pregnancy Pains See What They Do 📰 You Didnt Know These Top Parts Of A Shoe Are The Real Reason Your Feet HurtFinal 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:
\[
a_{10} = 3(10) + 2 = 30 + 2 = 32
\]
