Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

9. Sequences, Series, & Induction

Arithmetic Sequences

College Algebra

9. Sequences, Series, & Induction

Arithmetic Sequences

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2:15 minutes

Problem 25

Textbook Question

In Exercises 25–27, use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. The is a summation, refer to the textbook.

Verified step by step guidance

Identify the formula for the sum of the first n terms of an arithmetic sequence: \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( S_n \) is the sum of the first n terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.

Determine the values of \( a_1 \), \( a_n \), and \( n \) from the given problem or sequence.

Substitute the values of \( a_1 \), \( a_n \), and \( n \) into the formula \( S_n = \frac{n}{2} (a_1 + a_n) \).

Simplify the expression inside the parentheses: \( a_1 + a_n \).

Calculate \( \frac{n}{2} \times (a_1 + a_n) \) to find the sum of the first n terms.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. For example, in the sequence 2, 5, 8, 11, the common difference is 3. Understanding this concept is crucial for identifying the terms involved in the sum.

Sum of the First n Terms

The sum of the first n terms of an arithmetic sequence can be calculated using the formula S_n = n/2 * (a_1 + a_n), where S_n is the sum, n is the number of terms, a_1 is the first term, and a_n is the nth term. This formula allows for efficient calculation of the total sum without needing to add each term individually.

Formula Derivation

Deriving the formula for the sum of an arithmetic sequence involves recognizing that the sum can be expressed in two ways: forward and backward. By writing the sequence in reverse and adding the two equations, one can simplify to find the formula. This understanding helps in grasping why the formula works and how to apply it effectively.

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