Identify arithmetic and geometric series

Learning Objectives Define the difference between a sequence and a series. Define an arithmetic series and identify its common difference. Define a geometric series and identify its common ratio. Systematically test a series for a common difference. Systematically test a series for a common ratio. Accurately classify a given series as arithmetic, geometric, or neither. If you were offered a job that paid $1000 the first day, $2000 the second, $3000 the third, OR a job that paid $1 the first day, $2 the second, $4 the third, which would you choose? 🤔 The answer lies in identifying the pattern of the series! This tutorial will teach you how to distinguish between two fundamental types of series: arithmetic and geometric. Mastering this identification skill is the essential f...

TermDefinitionExample SeriesThe sum of the terms in a sequence. A series is represented by terms being added together.The sequence 2, 4, 6, 8 corresponds to the series 2 + 4 + 6 + 8. Term (a_n)An individual number or element in a series. The subscript 'n' denotes the term's position (e.g., a_1 is the first term).In the series 5 + 10 + 15 + 20, the third term (a_3) is 15. Arithmetic SeriesA series in which the difference between any two consecutive terms is constant. It represents adding a fixed amount repeatedly.3 + 7 + 11 + 15 + ... (Here, 4 is added to each term to get the next). Common Difference (d)The constant value that is added to each term to get the next term in an arithmetic series.In the series 10 + 8 + 6 + 4, the common difference is -2. Geometric SeriesA series...

Test for an Arithmetic Series A series is arithmetic if and only if `a_n - a_{n-1} = d` for all consecutive terms, where `d` is a constant. To check if a series is arithmetic, subtract each term from the term that follows it. If the result is the same for all pairs, the series is arithmetic and the result is the common difference, `d`. Test for a Geometric Series A series is geometric if and only if `a_n / a_{n-1} = r` for all consecutive terms, where `r` is a constant. To check if a series is geometric, divide each term by the term that precedes it. If the result is the same for all pairs, the series is geometric and the result is the common ratio, `r`.