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Sum of the First n Terms of an Arithmetic Sequence

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Sum of the First n Terms of an Arithmetic Sequence

Study Guide

Key Definition

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

Important Notes

The formula applies only to arithmetic sequences.
You need the first term and either the last term or the common difference to use the formula.
Alternatively, if you know the common difference $d$, you can use $S_n = \frac{n}{2} \cdot [2a_1 + (n - 1)d]$ without knowing the last term.
An arithmetic sequence increases or decreases by a constant amount.
The formula calculates what is known as an arithmetic series.
Ensure you have the correct number of terms $n$.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

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

$a^n$ represents exponentiation

$\sqrt{x}$ represents square root

Remember to use proper notation when solving problems

Why It Works

This formula works because it effectively averages the first and last term and multiplies by the number of terms, capturing the essence of an arithmetic sequence where each term adds a constant difference.

Remember

When calculating $S_n = \frac{n}{2} \cdot (a_1 + a_n)$, ensure you know both the first and last term or the common difference.

Quick Reference

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

Understanding Sum of the First n Terms of an Arithmetic Sequence

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Video explanation of this concept

Beginner

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Beginner Explanation

In an arithmetic sequence, the sum of the first $n$ terms can be found using $S_n = \frac{n}{2} \cdot (a_1 + a_n)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the first 5 terms of the arithmetic sequence where $a_1 = 2$ and $a_5 = 10$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you save $\$5$ on the first day, $\$10$ on the second day, and continue saving in this pattern. How much will you have saved by the end of 30 days?

Thinking Challenge

Thinking Exercise