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Sum of the First n Terms of a Series

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Sum of the First n Terms of a Series

Study Guide

Key Definition

The sum of the terms of a sequence is called a series. For arithmetic series, use $S_n = n \left( \frac{a_1 + a_n}{2} \right)$. For geometric series with first term $a_1$ and common ratio $r \neq 1$, use $S_n = a_1 \left( \frac{1 - r^n}{1 - r} \right)$.

Important Notes

The formula $S_n = n \left( \frac{a_1 + a_n}{2} \right)$ is for arithmetic series.
In a sequence, $a_1$ is the first term.
For geometric series, a different formula is used.
A sequence can be neither arithmetic nor geometric.
Use direct addition when other formulas don't apply.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

$\Sigma$ denotes summation

$S_n$ denotes the sum of the first n terms

$a_n$ denotes the nth term of a sequence

Remember to use proper notation when solving problems

Why It Works

The formula $S_n = n \left( \frac{a_1 + a_n}{2} \right)$ simplifies the process of summing terms in arithmetic sequences by using the first and last terms.

Remember

Using the formula $S_n = n \left( \frac{a_1 + a_n}{2} \right)$ helps avoid adding each term individually.

Quick Reference

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

Geometric Sum Formula:$S_n = a_1 \left( \frac{1 - r^n}{1 - r} \right)$

Understanding Sum of the First n Terms of a Series

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Video explanation of this concept

Beginner

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Beginner Explanation

An arithmetic series adds terms by a constant difference $d$. The nth term is $a_n = a_1 + (n - 1)d$, and the sum of the first n terms is $S_n = \frac{n}{2} (a_1 + a_n)$. For example, for 2 + 5 + 8 + 11 + 14, we have $a_1=2$, $a_5=14$, so $S_5 = \frac{5}{2} \times (2 + 14) = 40$.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the first 20 terms of an arithmetic series where $a_1 = 5$ and $a_{20} = 62$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You save $\$5$ every week and increase by $\$3$ each week. Calculate your savings after 10 weeks.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a geometric series with first term $a_1 = 3$ and common ratio $r = 2$. Find the sum of the first 5 terms.

Challenge Quiz

Single Choice Quiz

Advanced

If $a_1 = 2$ and the common difference $d = 4$, what is the sum of the first 25 terms?

Recap

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