Arithmetic progressions: definitions and examples

A sequence is a list of numbers $a_1, a_2, a_3, \ldots$ written in some specific order. The number $a_n$ is called the $n$th term of the sequence. Sequences can follow any rule , squares, primes, Fibonacci-like recurrences , but this chapter focuses on the simplest interesting rule: each term is obtained by adding a fixed number to the previous term.

Definitions

A sequence $a_1, a_2, a_3, \ldots$ is an arithmetic progression (AP) if there exists a number $d$ such that $a_{n+1} - a_n = d \quad \text{for every } n \ge 1.$

This $d$ is called the common difference. The first term is denoted $a$ (or $a_1$). An AP is completely specified by the pair $(a, d)$: $a, \quad a + d, \quad a + 2 d, \quad a + 3 d, \quad \ldots$

A finite AP has a fixed number of terms ($n$ terms). Its last term is denoted $\ell = a + (n - 1) d$.

A sequence is an AP iff every consecutive difference is the same. The test is: compute $a_2 - a_1$, $a_3 - a_2$, $a_4 - a_3$, etc. If they all match, it is an AP with that common value as $d$. If any differ, it is not.

Recognising APs

Examples and non-examples:

• $3, 7, 11, 15, 19, \ldots$ , differences are all $4$. AP, $a = 3, d = 4$.
• $5, 2, -1, -4, -7, \ldots$ , differences are $-3$ throughout. AP, $a = 5, d = -3$.
• $1, 4, 9, 16, 25, \ldots$ , differences are $3, 5, 7, 9$ (changing). Not an AP.
• $2, 4, 8, 16, 32, \ldots$ , differences are $2, 4, 8, 16$ (changing). Not an AP (this is a geometric progression).
• $-10, -10, -10, -10$ , common difference $0$. AP with $d = 0$ (a constant sequence).
• $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots$ , simplifying: $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \ldots$ , differences $\sqrt{2}$. AP.

Some real-life APs:

• The cost of a taxi: ₹$15$ for the first km, then ₹$8$ for each subsequent km , fares: $15, 23, 31, \ldots$. AP with $a = 15, d = 8$.
• Each row of an auditorium has $2$ more seats than the row in front. Seats per row form an AP.
• A salary scheme with a fixed annual increment.

Theorem / Concept: characterising APs

Test for an AP. A list $a_1, a_2, \ldots$ is an AP if and only if there is a constant $d$ with $a_{n+1} - a_n = d$ for all $n$. Equivalently, $2 a_{n+1} = a_n + a_{n+2}$ for all $n$ (the middle-term test).

The middle-term test is often easier in problems where you are given three numbers and asked to find a parameter that makes them an AP.

If the sequence is given by a formula (like $a_n = 3 n + 5$), you can read off $d$ by computing $a_{n+1} - a_n$. In general, $a_n$ is a linear function of $n$ , $a_n = a + (n - 1) d = d n + (a - d)$.

So an AP is exactly a sequence whose general term is linear in $n$. This is one of the cleanest characterisations and worth remembering.

Worked examples

Example 1. Find the first term and common difference of $7, 13, 19, 25, \ldots$.

$a = 7$. Differences: $13 - 7 = 6$, $19 - 13 = 6$, $25 - 19 = 6$. So $d = 6$.

Example 2. Is $-10, -6, -2, 2$ an AP? If so, find the next two terms.

Differences: $-6 - (-10) = 4$, $-2 - (-6) = 4$, $2 - (-2) = 4$. Yes, AP with $a = -10, d = 4$.

Next two terms: $2 + 4 = 6$ and $6 + 4 = 10$.

Example 3. For what value of $k$ are $k$, $2 k - 1$, $2 k + 1$ in AP?

Middle term test: $2(2 k - 1) = k + (2 k + 1) \Rightarrow 4 k - 2 = 3 k + 1 \Rightarrow k = 3$.

Example 4. The $n$th term of a sequence is $a_n = 3 n + 1$. Show it is an AP and find $a$ and $d$.

$a_n = 3 n + 1$ is linear in $n$. Differences: $a_{n+1} - a_n = 3(n+1) + 1 - (3 n + 1) = 3$. So AP with $d = 3$, and $a = a_1 = 4$.

Example 5. The first three terms of an AP are $2 x + 1, 3 x + 2, 2 x + 5$. Find $x$.

By the middle-term test: $2 (3 x + 2) = (2 x + 1) + (2 x + 5) \Rightarrow 6 x + 4 = 4 x + 6 \Rightarrow x = 1$.

Check: terms become $3, 5, 7$. Common difference $2$. ✓

Try it yourself

1. Which of these are APs? For each AP, find $a$ and $d$: (i) $11, 9, 7, 5, \ldots$, (ii) $2, 6, 12, 20, \ldots$, (iii) $0.5, 1, 1.5, 2, \ldots$, (iv) $\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots$, (v) $1, 1, 1, 1, \ldots$.
2. Find $x$ if $x - 2, 2 x, 3 x + 1$ are in AP.
3. Find the first term and common difference of an AP whose $5$th term is $19$ and $9$th term is $35$.
4. Write the first four terms of the AP with $a = -2, d = 0$.
5. Does the sequence $a_n = 4 - 2 n$ form an AP? If yes, find $d$.
6. Find $k$ so that $k + 9, 2 k - 1, 2 k + 7$ form an AP.
7. Are $1, 4, 9, 16$ an AP? Justify.
8. Find the $20$th term of $5, 12, 19, 26, \ldots$.
9. The $n$th term of an AP is $5 - 3 n$. Find $a$ and $d$.
10. Three consecutive terms of an AP are $a - 5, a, a + 5$. Verify by the test.

Pitfalls / Insight

• Always check the difference for the first pair AND a later pair. Sometimes only the first few terms look like an AP; later they don't.
• $d$ can be negative or zero. A constant sequence is a valid AP.
• Don't confuse AP with GP. Multiplying by a fixed factor each time is geometric, not arithmetic.

Insight. Identifying an AP is almost always the easy step. The real work is finding $a$ and $d$ from the information you have. Two pieces of information are typically enough , use them to set up two equations in $a, d$.