Math 1 · Unit 9 · Lesson 2 · part 2 of 5

Adding the same amount: the common difference

Adding gives arithmetic, multiplying gives geometric.

Adding the same amount: the common difference

Adding gives arithmetic, multiplying gives geometric. This lesson is the adding one. An arithmetic sequence is a list that adds the same amount to get each next term. That amount is the common difference, written d.

Find d: subtract

Take any term and subtract the one before it.

60, 68, 76, 84: 68 − 60 = 8, and 76 − 68 = 8. d = 8.

5, 9, 13, 17: d = 4.

d can be negative

40, 37, 34, 31: 37 − 40 = −3. d = −3.

The list goes down, because adding a negative number is taking away.

Always subtract in the same order: a term minus the one before it.

There are two ways to describe the list. Think of a tall building.

Recursive rule: the stairsTells you how to get the next term from the one before it. You climb one step at a time.

a₁ = 60, a_n = a_{n − 1} + 8

Say: a sub 1 is 60, and a sub n is a sub n minus 1, plus 8. It needs the starting term, or you have nowhere to stand.

Explicit rule: the elevatorTells you how to jump straight to any term from its term number.

a_n = 60 + 8(n − 1)

Say: a sub n is 60 plus 8 times n minus 1. Put in n = 20 and you are at row 20 without visiting rows 2 to 19.

a_n = a₁ + d(n − 1)

Why n − 1 and not n?To reach the 20th term you start at the 1st term and take 19 steps, not 20. The first term already used up one spot: you are standing on it before you take any step. So the number of steps is always one less than the term number. Theater, row 20: 60 + 8 × 19 = 60 + 152 = 212 seats.

Watch the steps. Each dot is one term, and the coral step between dots is the common difference.

Where to go next

Next: It is a line: slope and the zeroth termThe dots sit on a line, so an arithmetic sequence must be a line in disguise. All the partsArithmetic Sequences, Lines Made of Dots