Math 1 · Unit 9 · Lesson 2 · part 3 of 5
It is a line: slope and the zeroth term
The dots sit on a line, so an arithmetic sequence must be a line in disguise.
It is a line: slope and the zeroth term
The dots sit on a line, so an arithmetic sequence must be a line in disguise. Here is the connection.
- 1The common difference is the slope.Each step goes 1 to the right and d up: rise over run is d over 1. For the theater the slope is 8, which is 8 seats per row.
- 2The explicit rule simplifies to slope-intercept form.60 + 8(n − 1) = 60 + 8n − 8 = 8n + 52. Distribute the 8, then combine 60 and −8. That is y = mx + b with m = 8 and b = 52.
- 3The y-intercept is the zeroth term.52 is the term that would come before the first: one step backward from a1, 60 − 8 = 52. It sits at n = 0. There is no row 0 in the theater, so it is not part of the sequence, but it is where the line crosses the y-axis.
The trap: the first term is not the y-intercept. The first term is at n = 1, not n = 0. The y-intercept is one step before it: a1 − d. For 60, 68, 76 the first term is 60 but the y-intercept is 52.
| What it is | Function notation | Sequence notation | Say it |
|---|---|---|---|
| the term | f(n) | an | a sub n |
| the previous term | f(n − 1) | an − 1 | a sub n minus 1 |
| the next term | f(n + 1) | an + 1 | a sub n plus 1 |
| the first term | f(1) | a1 | a sub 1 |