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Sum of the First n Terms of an Arithmetic Series

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Sum of the First n Terms of an Arithmetic Series

Study Guide

Key Definition

The sum of the first n terms of an arithmetic series is given by the formula $S_n = \frac{n(a_1 + a_n)}{2}$ where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term.

Important Notes

An arithmetic series has a constant difference between consecutive terms.
The formula $S_n = \frac{n(a_1 + a_n)}{2}$ allows you to find the sum without adding each term individually.
Sigma notation $\sum$ is often used to express series.
Knowing the first and last terms of the series is crucial.
The formula applies only to arithmetic series.

Mathematical Notation

$\sum$ denotes a sum

$+$ represents addition

$\frac{a}{b}$ represents a fraction

$S_n$ denotes the sum of the first n terms

$a_1$ and $a_n$ represent the first and last terms respectively

$d$ denotes the common difference between consecutive terms

Remember to use proper notation when solving problems

Why It Works

The formula $S_n = \frac{n(a_1 + a_n)}{2}$ works because it calculates the average of the first and last terms, then multiplies by the number of terms to find the total sum.

Remember

Always identify the first and last terms, and ensure the series is arithmetic for the formula $S_n = \frac{n(a_1 + a_n)}{2}$ to be applicable.

Quick Reference

Sum Formula:$S_n = \frac{n(a_1 + a_n)}{2}$

Alternate Sum Formula:$S_n = \frac{n}{2}[2a_1 + (n - 1)d]$

Understanding Sum of the First n Terms of an Arithmetic Series

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Video explanation of this concept

Beginner

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Beginner Explanation

An arithmetic series is a list of numbers where each term increases by the same constant amount called the common difference, d. To find the sum of the first n terms, you take the average of the first term and the last term and then multiply by the number of terms. That gives the formula S_n = n(a_1 + a_n)/2.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the first 10 terms of the series $3 + 6 + 9 + \ldots$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you save $\$5$ in your first week, increasing by $\$2$ each subsequent week. How much will you have saved after 20 weeks?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the series $2, 5, 8, \ldots$. What is the sum of the first 40 terms?

Challenge Quiz

Single Choice Quiz

Advanced

Find the sum of the series $\sum_{k=1}^{50} (3k + 2)$.

Recap

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