Add the same number, again and again
A list where every term is the one before it plus a fixed number.
Reena's salary
Reena gets a job at ₹8000 a month. Every year her salary goes up by ₹500. Her salary in year 1, year 2, year 3, … is 8000, 8500, 9000, …
Each salary is made from the one before it. How?
What do you do to one year's salary to get the next year's salary?
What this lesson covers
The idea
An arithmetic progression (AP) is a list of numbers, called terms, in which each term except the first is obtained by adding a fixed number to the preceding term.
Reena's salary
Reena gets a job at ₹8000 a month. Every year her salary goes up by ₹500. Her salary in year 1, year 2, year 3, … is 8000, 8500, 9000, …
Each salary is made from the one before it. How?
What do you do to one year's salary to get the next year's salary?
- Add the same ₹500 every time
- Add a bigger raise every time
- Multiply by the same number every time
Run two machines
Each machine makes a list, one term at a time. The little window shows the number it adds. Press Next term and watch the window.
A fixed number
A list of numbers in which each term, except the first, is obtained by adding a fixed number to the term before it is called an arithmetic progression, or AP. Each number in the list is a term.
Reena's salary list is an AP: the machine added 500 every time. The squares 1, 4, 9, 16, … are not: it added 3, then 5, then 7. The number kept changing.
Notes
An AP is a list in which each term, except the first, is the term before it plus a fixed number.
Check yourself
Which list is made by adding the same number each time?
Shakila puts ₹100 in a money box on the first birthday, then ₹50 more each year: 100, 150, 200, … How many rupees does she put in on the 4th birthday?
Answer: 250
100, 150, 200, 250. Each amount is the one before it plus 50.
The pairs of rabbits at the start of each month are 1, 1, 2, 3, 5, 8, … Is this list made by adding a fixed number?
A machine starts at 20 and adds 7 each time. What is the 4th term?
Answer: 41
20, 27, 34, 41. The 4th term is 41.
- 3, 6, 9, 12, … — correct. Yes! It adds 3 every time: 3 + 3 = 6, 6 + 3 = 9, 9 + 3 = 12.
- 1, 2, 4, 7, …. Look at the numbers added: +1, +2, +3. They keep changing.
- 2, 4, 8, 16, …. Look at the numbers added: +2, +4, +8. They keep changing. (Each term is double the one before. That is a different pattern.)
- No. The numbers added are 0, 1, 1, 2, 3, which keep changing. — correct. Yes! 1 to 1 adds 0, 1 to 2 adds 1, 2 to 3 adds 1, 3 to 5 adds 2, 5 to 8 adds 3. Not a fixed number.
- Yes. It adds 1 each time.. Look at the first step: 1 to 1 adds 0, and 3 to 5 adds 2. It is not always 1.
- Yes. It adds 0 each time.. Only the first step adds 0. From 1 to 2 you add 1.