Arithmetic Sequences

Arithmetic sequences and the common difference

A sequence is an ordered list of numbers $a_1, a_2, a_3, \dots$ where $a_n$ is the term in position $n$. In some sequences you get from one term to the next by adding the same number every time. Those are the ones we study here.

Some problems call this an arithmetic progression. It is exactly the same thing. If $d > 0$ the terms increase, if $d < 0$ they decrease, and if $d = 0$ they never change.

To test whether a sequence is arithmetic, subtract each term from the next one. If every difference is the same number, it is arithmetic. If you have a formula for $a_n$, show that $a_{n+1} - a_n$ simplifies to a constant that does not depend on $n$.

The explicit formula

To reach the 100th term you do not want to add $d$ ninety-nine times. Look at the pattern: $a_2 = a_1 + d$, $a_3 = a_1 + 2d$, $a_4 = a_1 + 3d$. The number of $d$'s is always one less than the position, because from $a_1$ to $a_n$ you take $n - 1$ steps.

Finding $a_1$ and $d$ from other information

Often you are not given $a_1$ and $d$. You are given two terms, or one term and $d$. Use the formula $a_n = a_k + (n-k)d$ and count the jumps between the two positions.

You can also write one equation per known term, with $a_1$ and $d$ as unknowns, and solve the system by elimination. For the example: $a_1 + 2d = 17$ and $a_1 + 7d = -3$. Subtracting the first equation from the second gives $5d = -20$, the same equation as before. Counting jumps is just the elimination step done in your head. Use the system when it feels safer, especially with more complicated information.

Arithmetic means and consecutive terms

If $a$, $b$, $c$ are three consecutive terms of an arithmetic sequence, then $b - a = c - b$. Rearranging, $2b = a + c$, so the middle term is the average of its neighbors. That is why the middle term is called the arithmetic mean of the other two.

Inserting arithmetic means. To interpolate $k$ arithmetic means between $a$ and $b$ means to place $k$ numbers between them so that the whole list is arithmetic. The list has $k + 2$ numbers, so there are $k + 1$ jumps from $a$ to $b$:

Each jump is an equal share of the total distance $b - a$.

The sum of the first $n$ terms

How much is $1 + 2 + 3 + \dots + 100$? According to the famous story, young Carl Friedrich Gauss saw the trick right away: pair the first and last numbers ($1 + 100$), the second and next-to-last ($2 + 99$), and so on. Every pair adds to $101$, and there are $50$ pairs, so the sum is $5050$.

The same idea works for any arithmetic sequence. Write the sum forwards, then backwards, and add the two rows column by column.

Each column adds to $a_1 + a_n$ and there are $n$ columns, so $2S_n = n(a_1 + a_n)$. Dividing by $2$ gives the formula below.

Adding a block of terms

Sometimes you only want the terms from position $m$ to position $p$, not from the start. There are two ways to do it.

Method 1: apply the formula to the block. The block is itself an arithmetic sequence. Its first term is $a_m$, its last term is $a_p$, and it has $p - m + 1$ terms (count both ends!).

Method 2: subtract. The block is everything up to $p$ minus everything before $m$: $S_p - S_{m-1}$.

Solving for $n$

When a problem gives you the total $S_n$ and asks "how many terms?", the unknown is $n$. Substitute everything you know into $S_n$. The second form of the formula contains $n^2$, so you usually get a quadratic equation in $n$.

A solution that is negative or not a whole number is always thrown out. But sometimes both roots are positive integers, and then you have to look at the context.

Word problems

Stadium seats, bonuses that grow every month, training plans: ask yourself three questions.

1. Is the same amount added (or subtracted) each time? If so, it is arithmetic. Identify $a_1$ and $d$.

2. Do they ask about one position (use $a_n$) or about a total (use $S_n$)?

3. How many terms are there? Sometimes $n$ is given. Sometimes you must solve for it first, usually from a target value $a_n$.

Now go back to Emma. Her plan is arithmetic with $a_1 = 2$ and $d = 0.5$. On day 10 she runs $a_{10} = 2 + 9(0.5) = 6.5$ miles, and in the first 10 days she logs $S_{10} = \dfrac{10(2 + 6.5)}{2} = 42.5$ miles.

Harder modeling: angles and payments

In harder problems, the sequence is hidden in a geometry fact or a financial agreement, and the total $S_n$ is given. Use the total to find the missing piece.

When both ends of the sequence and the total are known, the sum formula is a linear equation in $n$, so there is no quadratic to solve. Once you have $n$, count jumps to find $d$. If $d$ turns out not to be a nice number, leave it as an exact fraction and round only at the end (for money, to the nearest cent).

Algebraic tricks and proofs

Three techniques show up in the hardest problems.

Trick 1: three numbers in an arithmetic sequence. Do not call them $a$, $a + d$, $a + 2d$. Call them $x - d$, $x$, $x + d$. The $d$'s cancel when you add, so the sum is simply $3x$ and you immediately know the middle number.

Trick 2: from the sum back to the terms. If you know a formula for $S_n$, then $a_n = S_n - S_{n-1}$ for $n \ge 2$, because $S_n$ is $S_{n-1}$ with one more term added. For the first term, $a_1 = S_1$ (the formula for $a_n$ must agree with it, so check).

Trick 3: proofs. To prove that three quantities $p, q, r$ are in an arithmetic sequence, show that $q - p = r - q$ (or $2q = p + r$). Start from the given hypothesis and use it to make the two differences equal.

Another useful idea is that when the hypothesis is $b^2 - a^2 = c^2 - b^2$, factoring each side as a difference of squares gives $(b - a)(b + a) = (c - b)(c + b)$. Combine this with common denominators when your target differences are fractions.

Summary