Arithmetic progressions

What you will be able to do

• find any term of an AP, counting from the start or from the end;
• find the sum of $n$ terms, in two forms;
• use the properties of an AP, including terms the same distance from the ends;
• choose unknown numbers in an AP so that the algebra stays short.

A constant step

The terms $5, 9, 13, 17, \dots$ of the last lesson rise by $4$ each time. A sequence that changes by the same amount at every step is an arithmetic progression, or AP.

In an AP, $a_{n+1} - a_n = d$ for every $n$. The number $d$ is the common difference, and $a = a_1$ is the first term.

After the first term there are $n - 1$ steps of $d$ to reach the $n$th term:

So $a_n$ is a linear expression in $n$. Plot the terms and they lie on a straight line with slope $d$. The reverse holds too: if $a_n = pn + q$, the sequence is an AP with $d = p$.

Counting from the end

An AP with $m$ terms ends at the last term $l$. Read backwards, it is an AP with first term $l$ and common difference $-d$. So the $n$th term from the end is

The sum of n terms

The terms the same distance from the two ends always add up to the same number. The first and last give $a + l$. The second and second last give $(a + d) + (l - d) = a + l$, and so on.

Write the sum forwards, then backwards, and add the two lines. Each of the $n$ columns gives $a + l$. So $2S_n = n(a + l)$, and

Use the first form when you know the last term, and the second when you know $d$.

Properties of an AP

Take the AP $a_1, a_2, a_3, \dots$ with common difference $d$, and a fixed number $k$.

• Add $k$ to every term. The result is an AP with the same difference $d$.
• Subtract $k$ from every term. Again an AP, with the same $d$.
• Multiply every term by $k$. An AP with difference $kd$.
• Divide every term by $k \neq 0$. An AP with difference $\dfrac{d}{k}$.

There is a test for $S_n$ as well. In an AP, $S_n = \dfrac{d}{2}n^2 + \left(a - \dfrac{d}{2}\right)n$. That is a quadratic in $n$ with no constant term. So $S_n = 2n^2 + 3n$ of the last lesson came from an AP, with $d = 4$.

Choosing unknown numbers

When a question gives the sum of a few numbers in an AP, choose them around the middle.

• Three numbers: $a - d$, $a$, $a + d$. Their sum is $3a$.
• Four numbers: $a - 3d$, $a - d$, $a + d$, $a + 3d$. Their sum is $4a$, and the common difference is $2d$.

The $d$ cancels in the sum, so the sum alone gives $a$ at once.

A worked example

The $4$th term of an AP is $11$, and its $9$th term is $26$. Find the sum of its first $20$ terms.

1. Two equations. $a + 3d = 11$ and $a + 8d = 26$.
2. Subtract. $5d = 15$, so $d = 3$. Then $a = 11 - 9 = 2$.
3. The sum. $S_{20} = \dfrac{20}{2}\left(2 \times 2 + 19 \times 3\right) = 10 \times 61 = 610$.

Practise with new numbers

Try it in Python

The program lists an AP, finds its sum three ways, and adds the terms in pairs from the two ends.

Common mistakes

• Writing $a + nd$ for the $n$th term. There are $n - 1$ steps after the first term.
• Taking the wrong sign for $d$. In $40, 37, 34, \dots$, $d = -3$, not $3$.
• Forgetting that four numbers in an AP step by $2d$. With $a - 3d, a - d, a + d, a + 3d$, the difference is $2d$.
• Rejecting a negative answer too soon. A term, a difference or even a sum can be negative.

Remember these

• $a_n = a + (n - 1)d$, and the $n$th term from the end is $l - (n - 1)d$.
• $S_n = \dfrac{n}{2}(a + l) = \dfrac{n}{2}\left(2a + (n - 1)d\right)$.
• Adding, subtracting, multiplying or dividing every term by a constant keeps an AP.
• Terms the same distance from the two ends add up to $a + l$.
• Three numbers: $a - d, a, a + d$. Four numbers: $a - 3d, a - d, a + d, a + 3d$.

Check yourself

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

1. Find the $30$th term of the AP $8, 5, 2, \dots$.

2. An AP has $a_n = 7n - 4$. What are its first term and common difference?

3. Find the sum of the AP $3 + 8 + 13 + \dots + 93$.

4. Every term of an AP with $d = 4$ is multiplied by $-2$. Is the result an AP? What is its common difference?

5. Three numbers in an AP add up to $24$. What is the middle one?