Sum of first 10 terms: $ \frac102 \times (5 + 32) = 5 \times 37 = 185 $

Understanding the Sum of the First 10 Terms Using the Arithmetic Series Formula

Calculating the sum of a sequence is a fundamental concept in mathematics, especially when working with arithmetic series. One interesting example involves computing the sum of the first 10 terms of a specific series using a well-known formula — and it aligns perfectly with $ \frac{10}{2} \ imes (5 + 32) = 5 \ imes 37 = 185 $. In this article, we’ll explore how this formula works, why it’s effective, and how you can apply it to solve similar problems efficiently.

Understanding the Context

What Is an Arithmetic Series?

An arithmetic series is the sum of the terms of an arithmetic sequence — a sequence in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.

For example, consider the sequence:
$ a, a + d, a + 2d, a + 3d, \dots $
where:
- $ a $ is the first term,
- $ d $ is the common difference,
- $ n $ is the number of terms.

The sum of the first $ n $ terms of such a series is given by the formula:

$The sum of first 10 terms of the series \( \left(x+\frac{1}{x}\right ...$

Image Gallery

$The sum of first 10 terms of the series \( \left(x+\frac{1}{x}\right ...$

$calculus - First four terms of $ \sum_{k=1}^\infty \frac{(-1)^{k+1(2^k ...$

SOLVED:Show that ∑k=3^9 (k)/(k+1) is equal to ∑k=4^10 (k-1)/(k) by ...

10 Times Table Worksheet [10 Multiplication Table] Free PDF

Free Times Tables Printable: Fun Math PDFs - Printables for Everyone

Key Insights

\[
S_n = \frac{n}{2} \ imes (2a + (n - 1)d)
\]

Alternatively, it can also be written using the average of the first and last term:

\[
S_n = \frac{n}{2} \ imes (a + l)
\]

where $ l $ is the last term, and $ l = a + (n - 1)d $.

🔗 Related Articles You Might Like:

📰 code etruesports 📰 code geass characters 📰 code lyoko code 📰 From Console To 3D Marvel Wii Us Ultimate Mario Overhaulwatch It Go Viral 📰 From Copious Honey To Heartwarming Chaoswatch What Winnie The Pooh And Friends Did 📰 From Corn To Crumbs Discover The Complete Turkey Diet Revealed 📰 From Corona To Canvas The Powerful Wirehaired Pointers Must Watch Hunting Moments 📰 From Cosplay To Controversy The Rise Of White Power Ranger And Its Influence 📰 From Cowboys To Captions The Most Stylish Western Font Thats Going Viral 📰 From Cozy Firelit Tables To Gourmet Winter Feasts Bold Dinner Ideas To Sizzle This Season 📰 From Cozy To Chic Perfect Winter Outfits Every Woman Should Own This Season 📰 From Crickets To Leaf Lizardswhat Do Real Geckos Eat The Truth Exploded 📰 From Crispy Wings To Jammin Sauce This Wing Mixture Is Oversaturating Foodie Spaces 📰 From Curious To Confident What A Diana Piercing Really Does To Your Style 📰 From Curious To Obsessed Discover The Magic Of Whipped Cream Cheese Frosting 📰 From Cuteness To Chilling What Makes A Yandere So Dangerously Fanatical 📰 From Day One Weep Holes Make You Cryheres Why Theyre Irresistible 📰 From Day To Night The Elegant White Slip Dress Every Fashionista Is Wearing

Final Thoughts

Applying the Formula to the Given Example

In the expression:
\[
\frac{10}{2} \ imes (5 + 32) = 5 \ imes 37 = 185
\]
we recognize this as a concise application of the arithmetic series sum formula.

Let’s match the terms to our general strategy:

$ n = 10 $ — we want the sum of the first 10 terms
- First term, $ a = 5 $
- Last term, $ l = 32 $ (which is $ 5 + (10 - 1) \ imes d = 5 + 9 \ imes d $. Since $ d = 3 $, $ 5 + 27 = 32 $)

Now compute:
\[
S_{10} = \frac{10}{2} \ imes (5 + 32) = 5 \ imes 37 = 185
\]

Why This Formula Works

The formula leverages symmetry in the arithmetic series: pairing the first and last terms, the second and second-to-last, and so on until the middle. Each pair averages to the overall average of the sequence, $ \frac{a + l}{2} $, and there are $ \frac{n}{2} $ such pairs when $ n $ is even (or $ \frac{n - 1}{2} $ pairs plus the middle term when $ n $ is odd — but not needed here).

Thus,