Arithmetic sequences

Learning Objectives Define an arithmetic sequence and identify its key components. Determine the common difference of an arithmetic sequence. Use the explicit formula to find the value of any term (the nth term) in an arithmetic sequence. Determine the position (n) of a given term in an arithmetic sequence. Find the first term or common difference given two non-consecutive terms. Model and solve real-world problems using arithmetic sequences. Imagine you start a new savings plan, putting away $10 the first week and increasing the amount by $5 each week. How much will you save in the final week of the year? 🤔 This tutorial introduces arithmetic sequences, which are ordered lists of numbers with a constant pattern of addition or subtraction. Understanding these sequences is...

TermDefinitionExample SequenceAn ordered list of numbers, often following a specific pattern or rule.2, 4, 6, 8, 10, ... TermAn individual number or element within a sequence. We use subscript notation like a_n to denote the term in the nth position.In the sequence 2, 4, 6, 8, ..., the third term (a_3) is 6. Arithmetic SequenceA sequence in which the difference between any two consecutive terms is constant.5, 9, 13, 17, ... (The difference is always +4). Common Difference (d)The constant value that is added to each term to get the next term in an arithmetic sequence. It can be positive or negative.In the sequence 30, 25, 20, 15, ..., the common difference (d) is -5. First Term (a_1)The very first number in a sequence.In the sequence 7, 14, 21, 28, ..., the first term (a_1) is 7. nth Term...

Common Difference Formula d = a_n - a_{n-1} To find the common difference (d), subtract any term from the term that immediately follows it. For example, d = a_2 - a_1. The nth Term Formula (Explicit Formula) a_n = a_1 + (n - 1)d This is the most important formula for arithmetic sequences. It allows you to find the value of any term (a_n) if you know the first term (a_1), the term's position (n), and the common difference (d).