AI Tutor
Every time you stream video on your phone, your battery drops by exactly 4%4\% an hour. If you start at 88%88\%, what is your battery after one hour, two hours, or three hours?
This predictable chain of numbers is an arithmetic sequence — an ordered list of numbers where the difference between consecutive terms is always constant.
Can we describe this step-by-step jump using simple math notation?

The Step-by-Step Pattern

In sequence notation, we label each number by its position: a1a_1 is the first term, a2a_2 is the second term, and ana_n is the nthn^{\text{th}} term. The fixed amount added each step is the common difference, written as dd.
If every term simply takes the term right before it and adds dd, how do we capture that relationship in an algebraic rule?

Building the Recursive Formula

A recursive formula is a rule that builds each new term using the term that came right before it. To write one, you need two essential parts: a starting point and a transition rule.
If ana_n represents the current term, then an−1a_{n-1} represents the previous term (the term at index n−1n - 1). Adding the constant change dd gives the recurrence relation: an=an−1+da_n = a_{n-1} + d for n≥2n \ge 2.
What happens if someone gives you the transition rule an=an−1+5a_n = a_{n-1} + 5, but forgets to tell you where to start?

Why the Starting Point Matters