Every AP looks like this
Build the terms from a first block a and steps of d.
Build the third term
Start an AP at a and add a fixed number d each time. The 2nd term is a + d.
The 3rd term is the 2nd term with another d added.
What is the 3rd term?
What this lesson covers
The idea
An AP with first term a and common difference d has the general form a, a + d, a + 2d, a + 3d, …
Build the third term
Start an AP at a and add a fixed number d each time. The 2nd term is a + d.
The 3rd term is the 2nd term with another d added.
What is the 3rd term?
- a + 2d
- a + 3d
- 2a + d
Build the terms from blocks
The big block is a. Each small block is one d. Press Next term: the new row is the row above with one more d.
The general form
An AP with first term a and common difference d is a, a + d, a + 2d, a + 3d, … This is called its general form.
Look at the blocks: the 2nd term has one d, the 3rd has two d's, the 4th has three. Put any numbers in place of a and d, and the same blocks make the list.
Notes
a, a + d, a + 2d, a + 3d, … is the general form of an AP with first term a and common difference d.
Check yourself
What is the 5th term of the AP with first term a and common difference d?
For a = 7 and d = 3 the first terms are 7, 10, 13, … What is the 4th term, a + 3d?
Answer: 16
a + 3d = 7 + 3 × 3 = 7 + 9 = 16.
An AP has a = 20 and d = −4. What is its 3rd term, a + 2d?
Answer: 12
a + 2d = 20 + 2 × (−4) = 20 − 8 = 12.
Which list has the general form a, a + d, a + 2d, … with a = 4 and d = 5?
- a + 4d — correct. Yes! The 2nd term is a + d, the 3rd a + 2d, the 4th a + 3d, so the 5th is a + 4d.
- a + 5d. That is the 6th term. The list goes a, a + d, a + 2d, a + 3d, a + 4d, …
- 5a + d. a is only there once. The terms are a, a + d, a + 2d, …
- 4, 9, 14, 19, … — correct. Yes! 4, then 4 + 5 = 9, then 4 + 2 × 5 = 14, then 4 + 3 × 5 = 19.
- 5, 9, 14, 19, …. It must start at a = 4.
- 4, 5, 6, 7, …. That list adds 1 each time, so d = 1. Here d = 5.