AI Tutor
Imagine a summer job that pays you 15onday1,15 on day 1, 19 on day 2, and $23 on day 3.
Adding $4 each day is simple enough for the first week, but what if you want to know your exact pay on day 100 without adding 99 times?
To jump straight to any term without listing every single value along the way, we need a mathematical shortcut.

The Step Pattern

In an arithmetic sequence, the difference between consecutive terms is always constant. We call this step size the common difference (dd), while the very first term is written as a1a_1.
Notice how many times we add dd: to reach term 2, we add dd once; to reach term 3, we add dd twice; to reach term 4, we add dd three times.
Because you are already standing on the first term at the start, reaching the nn-th term requires exactly n−1n - 1 steps of size dd.
How can we package this simple counting logic into one universal rule?

The Explicit Formula

An explicit formula is a rule that computes any term ana_n directly from its position number nn.
Mathematically, since a2=a1+da_2 = a_1 + d, a3=a1+2da_3 = a_1 + 2d, and a4=a1+3da_4 = a_1 + 3d, following this exact chain produces the general rule an=a1+(n−1)da_n = a_1 + (n - 1)d.