What is an arithmetic progression?

Patterns that grow by steps

Look at these lists of numbers from everyday life.

• A shop charges ₹30 to deliver a parcel of up to $1\ \text{kg}$, and ₹8 more for each extra kilogram: $30, 38, 46, 54, \dots$ rupees.
• A candle is $20\ \text{cm}$ tall and burns down $1.5\ \text{cm}$ every hour: $20, 18.5, 17, 15.5, \dots$ centimetres.
• Meena puts ₹25 in her money box every Sunday. Her savings after each Sunday are $25, 50, 75, 100, \dots$ rupees.
• A cinema ticket costs ₹150 for every show of the day: $150, 150, 150, \dots$ rupees.

In each list, you get the next number by adding the same fixed amount to the one before it. In the candle list, that amount is $-1.5$: adding a negative number makes the list go down. In the last list it is $0$.

The definition

An arithmetic progression (AP) is a list of numbers in which each term, except the first, is obtained by adding a fixed number to the term before it.

• The numbers in the list are its terms: $a_1, a_2, a_3, \dots$
• The first term is called $a$, so $a = a_1$.
• The fixed number is the common difference, written $d$.

So the terms of every AP are

This is the general form of an AP.

The common difference is any term minus the term just before it:

An AP may stop after a number of terms, like the pages of a book, $1, 2, 3, \dots, 240$. It is then a finite AP and has a last term. An AP that goes on for ever, like $5, 10, 15, \dots$, is infinite.

Is this list an AP?

To check a list, work out every difference between neighbouring terms. If all of them are equal, the list is an AP. One different difference is enough to say no.

A worked example. Which of these are APs?

1. $7, 3, -1, -5, \dots$ The differences are $3 - 7 = -4$, then $-1 - 3 = -4$, then $-5 - (-1) = -4$. All equal: an AP with $d = -4$.
2. $2, 6, 18, 54, \dots$ The differences are $4$, $12$ and $36$. They are not equal, so it is not an AP. Each term is three times the one before, which is a different kind of pattern.
3. $\dfrac{1}{2}, \dfrac{5}{6}, \dfrac{7}{6}, \dfrac{3}{2}, \dots$ Write them with denominator $6$: $\dfrac{3}{6}, \dfrac{5}{6}, \dfrac{7}{6}, \dfrac{9}{6}$. Each difference is $\dfrac{2}{6} = \dfrac{1}{3}$. An AP.

Writing an AP from $a$ and $d$. With $a = -3$ and $d = 5$, the AP is $-3, 2, 7, 12, \dots$

Practise with new numbers

Try it in Python

A computer can check every difference in a moment. This program tests four lists and says which ones are APs.

Common mistakes

• Subtracting the wrong way. $d$ is a term minus the term before it. For $15, 11, 7$, $d = 11 - 15 = -4$, not $4$.
• Checking only the first two terms. In $1, 3, 5, 9$ the first differences are $2$ and $2$, but the last is $4$. It is not an AP.
• Mixing up adding and multiplying. In $2, 6, 18$ each term is multiplied by $3$. That is not a common difference.
• Thinking $d$ cannot be $0$ or negative. Both are allowed: $9, 9, 9$ and $9, 6, 3$ are APs.

Remember these

• An AP adds the same number $d$ each time: $a, a + d, a + 2d, \dots$
• $d =$ any term minus the term just before it. It can be positive, negative or zero.
• To test a list, every difference between neighbours must be the same.
• A finite AP has a last term. An infinite AP goes on for ever.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Is $3, 3.5, 4, 4.5, \dots$ an AP? If it is, give $d$.

2. Write the first four terms of the AP with $a = 10$ and $d = -3$.

3. Is $1, 2, 4, 8, 16$ an AP?

4. Pooja says the common difference of $4, 1, -2$ is $3$. Is she right?

5. Can every term of an AP be the same number?