Sequence & Series Basics

Sequences map natural numbers to real values, exploring discrete accumulation. While arithmetic series exhibit unbounded linear divergence, geometric series reveal conditional convergence bounds. An absolute ratio strictly below unity is the mathematical prerequisite for finite infinite-summability.

Both sequence types start from a first term, but they grow by different rules. An arithmetic sequence adds the same amount each step (like saving $50 every month), while a geometric sequence multiplies by the same factor (like a balance earning 5% interest yearly). The nth-term formula hands you any single value without listing them all, and the partial sum adds up the first n. Watch how one starting value diverges wildly depending on which rule you pick.

Try to build a sequence that is simultaneously arithmetic and geometric. It turns out only one family works: a constant sequence like 7, 7, 7, …, where the difference is d = 0 and the ratio is r = 1. Any genuine change breaks one rule or the other, because steady adding always curves away from steady multiplying. It is a rare case where two seemingly independent growth laws are forced to collapse into a single, motionless value.