Arithmetic and geometric sequences

Sequences

A sequence is a list of numbers that follows a pattern from one term to the next. A sequence usually continues forever, repeating the same rule each time you move to the next number.

Two common types of sequences are:

• Arithmetic sequences, which follow a pattern of adding the same amount each time
• Geometric sequences, which follow a pattern of multiplying by the same amount each time

In this section, you’ll mainly use two approaches:

• Expand the sequence by hand (write out terms until you reach the one you need)
• Use a formula that matches the type of sequence

We’ll also use some vocabulary throughout this section:

• A term is a number in the sequence.
• The first number is the first term.
• The letter $n$ represents the position of a term in the sequence.
• The phrase $n$th term means “the term in position $n$” (it could be the 1st, 2nd, 3rd, and so on).

Arithmetic sequences

An arithmetic sequence is a sequence where you add the same number each time you move to the next term. That constant amount is called the common difference.

This means the difference between any two consecutive terms is the same. For example, the amount from the 1st term to the 2nd term is the same as the amount from the 2nd term to the 3rd term.

Let’s show that idea on a number line:

This pattern continues forever, so you can:

• count forward term by term until you reach the one you want, or
• use a formula to jump directly to the $n$th term.

The $n$th term of an arithmetic sequence is:

x_n = (\text) + \text * (n-1)

The ACT won’t require a specific method. Use whichever is faster and clearer for you.

For example:

Find the $9$th term in the arithmetic sequence that begins as follows: $1,5,9,...$

Here are two reliable methods:

1. Write out the terms until you reach the one you need. (The pattern is adding four, so it continues $1,5,9,13,17,21,25,29,33$. The $9$th term in the sequence is $33$)

2. Use the formula to find the term directly: $x=1+4\ast (9-1)$, $x=33$

The key idea to remember: arithmetic sequences are number lists with an additive pattern.

Geometric sequences

Geometric sequences are similar to arithmetic sequences in that they follow a consistent pattern from term to term. The difference is the type of pattern:

• Arithmetic sequences add a constant amount.
• Geometric sequences multiply by a constant amount.

That constant multiplying factor is often called the ratio. In a geometric sequence, the ratio between any two consecutive terms is the same.

Compare this idea to the number line below:

As with arithmetic sequences, this pattern continues forever. You can always find a term by listing values, but you can also use a formula.

The $n$th term of a geometric sequence is:

x_n = (\text) * (\text)^NaN

For example, if the multiplying ratio is $4$ and the first term is $1$, the sequence is:

$1,4,16,64,256,1024,4096,16384,...$

Geometric sequences can grow much faster than arithmetic sequences, but the structure is the same: each step uses the same rule.

Summing sequences

Sometimes you won’t be asked for a single term. Instead, you’ll be asked for the sum of the terms up to a certain point.

The simplest method is often to:

• write out the terms you need, then
• add them.

However, some problems won’t give you enough information to list every term easily. In those cases, you’ll use a sum formula.

In these more advanced cases, refer to the following formulas, which sum up the series to a given term:

Definitions

Sum of an arithmetic sequence

$n/2\ast (2a+d\ast (n-1))$ where $a=$ first term, $d=$ difference (or number by which each term increases), and $n=$ number of terms.

Sum of a geometric sequence

$a\ast ((1-r^n)/(1-r))$ where $a=$ first term, $r=$ ratio (or number by which each term is multiplied), and $n=$ number of terms.

Let’s review an example using the simplest approach (list the terms, then add):

What is the sum of a geometric sequence with five terms where the first term is $4$ and the multiplying ratio is $3$?

Start by listing the terms. You begin at $4$ and multiply by $3$ each time until you have five terms:

$4,12,36,108,324,...$

Now add the first five terms.

Listing and adding is often the quickest method. If you can’t list the terms (or if the problem gives you the sum and asks you to work backward), use the formulas above.

Key points

Sequences. Sequences are lists of numbers that follow a pattern.

Arithmetic sequences. Arithmetic sequences are ones that follow an additive pattern as the terms increase.

Geometric sequences. Geometric sequences are ones that follow a multiplicative pattern as the terms increase.

Methods of solution. Solving arithmetic sequences is equally effective whether done by using common sense of patterns or using the formula.

Sums of sequences. Add up all of the terms until the one you need, or use the formulas discussed above for each sequence.