Arithmetic Sequences

Picture a staircase where every step is exactly the same height. That's an arithmetic sequence: start with some value, then keep adding the same number, called the common difference, over and over. Because that number never changes, each term grows by the same fixed jump, so if you plot term number on the x-axis and term value on the y-axis, the points line up perfectly straight. The slope of that line is the common difference itself.

You don't have to add one step at a time to get far down the staircase. If a1 is the first term and d is the common difference, the nth term is a_n = a1 + (n-1)d. That formula just counts how many jumps you need: to go from term 1 to term n takes exactly n-1 jumps of size d, so you add d that many times to the start. This is the direct shortcut from any known term to any other: whatever term you know, count the difference in index to the term you want, multiply by d, and add or subtract from the known value.

This idea sets up the whole unit. An arithmetic sequence grows by repeated addition, a straight-line pattern. Later in the unit, geometric sequences grow by repeated multiplication, and that bend from a straight line into a curve is exactly what separates linear growth from exponential growth, and exponential growth from its mirror image, the logarithm.