AI Tutor
Legend says that when ten-year-old Carl Friedrich Gauss was told to add every number from 11 to 100100 as busywork, he wrote the answer down in seconds.
He didn't add them one by one; he spotted a symmetry hiding inside constant-step patterns.

Anatomy of an Arithmetic Sequence

An arithmetic sequence is an ordered list of numbers where each term after the first is found by adding a fixed value called the common difference (dd).
To find dd, subtract any term from the term that comes immediately after it: d=an−an−1d = a_n - a_{n-1}.
How do we write rules to describe these sequences step-by-step versus jumping straight to any term?

Recursive and Explicit Formulas

A recursive formula builds each term from the previous one, stating a1a_1 as the starting value and an=an−1+da_n = a_{n-1} + d for n≥2n \ge 2.
An explicit formula lets you jump straight to the nn-th term without calculating all the terms before it: an=a1+(n−1)da_n = a_1 + (n - 1)d.
Why is it (n−1)(n - 1)? Starting at a1a_1, you take 00 steps; to reach a2a_2 you take 11 step of dd; so reaching ana_n requires taking n−1n - 1 steps of size dd.
Let's put this into practice on a sequence with negative numbers and fractions.