Patterns Examples

Write down the formula for the general term in the sequence

2, 5, 10, 17, 26,...

starting with n = 1.

This sequence is very close to the sequence of squares. If we re-write each term we can see how n affects the value of the term

The formula for the general term is

a_n = n² + 1.

Find a formula for the general term of the sequence

0, 2, 6, 12, 20,...

starting with n = 1.

This looks like it's growing at roughly the same rate as the sequence of squares. Let's break the terms down into a square plus or minus something and see what we get.

The formula for the general term is

a_n = n² – n.

Before starting the exercises, be aware that there can be multiple definitions of the same sequence. For example, the sequence

2, 5, 8, 11,...

can be defined by either

a_n = 2 + 3(n – 1)

or

a_n = 3n – 1.

If you get the correct terms when you evaluate your formula, your answer is correct, whether it agrees with ours or not. To check your answers when you're not sure if you found the correct formula, evaluate the first 3 or 4 terms using your formula, and see if you get the terms that were given.

While doing the exercises, make sure you get an answer of your own before looking at ours.

There's a big difference between recognizing that an answer is correct and being able to come up with your own answer.

In order to do well on homework, tests, and life, you need to be able to come up with your own answers, not just recognize when someone else finds one.

Find a formula for the general term of the sequence

starting with n = 1.

Let's do the easy part first. Look at the denominators of the terms:

The denominator of the n^th term is n!. Now look at the numerators, which we can rewrite as powers of 2:

The numerator of the n^th term is 2ⁿ. The formula for the general term is

Sometimes one part of the formula seems shifted, so we need an n in one part of the formula but (n + 1) in another part. Tricky. This can be affected by whether we start the sequence at n = 0 or n = 1.

Find a formula for the general term of the sequence

starting with n = 1.

The numerators can be rewritten as powers of 2:

The numerator of the general term is 2ⁿ. The denominators are factorials, but starting at 2! instead of 1!.

Let's make a table to help us see the relationship between n and the denominator.

The denominator of the n^th term is (n + 1)!. The formula for the general term is

Find a formula for the general term of the sequence

2, -4, 8, -16, 32,....

starting at n = 1.

First, ignore the signs. The sequence

2, 4, 8, 16, 32,...

is the sequence of powers of 2, with the n^th term given by 2ⁿ.

In order to make the signs alternate we need to multiply 2ⁿ by either

(-1)ⁿ or (-1)^{n + 1}. If we multiply by (-1)ⁿ the first term will be

a₁ = (-1)¹2¹ = -2.

This has the wrong sign, so we need to multiply by (-1)^{n + 1} instead. Then the first term is

a₁ = (-1)^{1 + 1}2¹ = 2,

just like it should be. The formula for the general term is

a_n = (-1)^{n + 1}2ⁿ.

We could also use

a_n = (-1)^{n – 1}2ⁿ.

Find a formula for the general term of the sequence

starting at n = 1.

First ignore the signs and find the general term for the sequence

The numerator of the n^th term is n. The denominators are powers of 2, but starting at 2² instead of 2¹.

The general term of this sequence, ignoring the signs, is

To make the signs alternate and have the first term be positive, we need to multiply by (-1)^{n + 1}. The general term of the original sequence, accounting for the sign changes, is

Find a formula for the general term of the sequence

starting at n = 1.

Ignore the signs. The denominator of the n^th term is n!. The numerator of the n^th term is (2n – 1), since the numerators are odd numbers starting at 1.

Now pay attention to the signs again. Since the signs alternate and the first term is negative, we also need a factor of (-1)ⁿ. The formula for the general term is