# Sequences

A sequence is a list of numbers that may be generated by some rule. In this lesson, we will study several different kinds of sequences.

## Arithmetic sequences

In the sequence of numbers

$$3,5,7,9,...$$

each number is 2 larger than the previous one.

 What is the 3rd number in this sequence?
 What number follows 9 in this sequence?

We will call the $n$th number in this sequence $\cl"red"{A(n)}$. For example, $A(2)=5$ because 5 is the 2nd number in the sequence.

 What is $\cl"red"{A(4)}$?
 What is $\cl"red"{A(6)}$?

A sequence where you can get each number by adding the same amount to the previous number is called an arithmetic sequence (pronounced “arithMETic”).

The points $\cl"red"{(n,A(n))}$ are plotted on the grid to the left. Notice that they all seem to lie on the same line. We’ll see if we can find a linear formula for $\cl"red"{A(n)}$ in terms of $n$.

In the table below, fill in the numbers of this sequence $\cl"red"{A(n)}$, as well as the value of the expression $2n+1$, for each value of $n$.

$n$$\cl"red"{A(n)}$$2n+1$
 Are $\cl"red"{A(n)}$ and $2n+1$ equal for each $n$ in this table?

In fact, the $n$th number in the sequence, $\cl"red"{A(n)}$, is always equal to $2n+1$. This gives us a formula for the sequence, which we can use to calculate any number in the sequence without already knowing all the ones before it. For example, $A(50)=2(50)+1=101$.

What is the 100th number in the sequence?

Compute the quantities $A(n+1)-A(n)$ in the table below.

$n$$A(n)$$A(n+1)$$A(n+1)-A(n) Because A(n+1) is always 2 larger than A(n), the difference A(n+1)-A(n) will always be equal to 2. This gives you a way to check whether a sequence is arithmetic: If a sequence S(n) is arithmetic, then S(n+1)-S(n) will always be the same no matter what n is. The number S(n+1)-S(n) is called the common difference of the arithmetic sequence S(n). Notice that this is similar to a formula for the slope m of a linear graph: m = f(x+1) - f(x). Now we’ll look at a different sequence of numbers:$$ 7,4,1,-2,... $$In this sequence, each number is 3 smaller than the previous one. Another way to say this is that each number is -3 larger than the previous one. So this is another arithmetic sequence.  What is the 5th number in this sequence? We will call the nth number in this sequence \cl"red"{B(n)}. For example, B(2)=4 because 4 is the 2nd number in the sequence.  What is \cl"red"{B(6)}?  What is the common difference of this sequence? (That is, B(2)-B(1), or B(3)-B(2), or ...) Again, the points \cl"red"{(n,B(n))} on the grid to the left all seem to fall on a line, so we’ll try to compute them using a linear expression. In the table below, fill in the numbers of this sequence \cl"red"{B(n)}, as well as the value of the expression -3n+10, for each value of n. n$$\cl"red"{B(n)}$$-3n+10  Are \cl"red"{B(n)} and -3n+10 equal for each n in this table? As before, \cl"red"{B(n)} is always equal to -3n+10. What is the 100th number in this sequence? Notice that we have found linear formulas (that is, formulas that look like mx+b) for both \cl"red"{A(n)} and \cl"red"{B(n)}. In fact: In an arithmetic sequence where the common difference is d, the nth number can be given by a linear formula of the form dn+b for some number b. Let’s look at the arithmetic sequence C(n) which begins$$ 2,5,8,11,... $$Notice that C(1)=2, and the common difference of this sequence is 3. Another way to say this is that the sequence is defined by:$$ C(n)=\{ \table 2, \text"if"\;n=1; C(n-1) + 3, \text"if"\;n>1 $$Because the common difference is 3, there must also be a formula for C(n) which looks like C(n)=3n+b for some number b. This formula tells us that C(1)=3(1)+b=3+b.  If C(1)=3+b and C(1)=2, then 3+b=2. What is the solution to the equation 3+b=2?  What is a simple formula for C(n)? ## Geometric sequences Now we’ll look at the sequence$$ 6,12,24,48,... $$where each number is 2 times the previous number. Call the nth number in this sequence \cl"red"{G(n)}.  The fifth number in the sequence, G(5), is 2(48)=96. What is G(6)? (Use a calculator if necessary.) A sequence where you can get each number by multiplying the previous number by the same amount is called a geometric sequence. The first six points \cl"red"{(n, G(n))} have been plotted on the grid to the left.  When we plotted the points in the arithmetic sequences \cl"red"{A(n)} and \cl"red"{B(n)}, they formed a straight line. Do the points in the geometric sequence \cl"red"{G(n)} form a straight line? Because the points in the geometric sequence \cl"red"{G(n)} don’t form a straight line, we’ll have to look for some formula that isn’t linear to describe them. In the table below, fill in the numbers of this sequence \cl"red"{G(n)}, as well as the value of the expression 3 ⋅ 2^n for each value of n. n$$\cl"red"{G(n)}$$3 ⋅ 2^n Compute the quantities$${G(n+1)}/{G(n)}$$in the table below. n$$G(n)$$G(n+1)$$${G(n+1)}/{G(n)}$$Because G(n+1) is always 2 times G(n), the quotient$${G(n+1)}/{G(n)}$$will always be equal to 2. This gives you a way to check whether a sequence is geometric: If a sequence S(n) is geometric, then$${S(n+1)}/{S(n)}$$will always be the same no matter what n is. The number$${S(n+1)}/{S(n)}$$is called the common ratio of the geometric sequence S(n). Now let’s look at another geometric sequence:$$ 18,-6,2,-2/3,... $$where each number is$$-1/3$$times the previous number (that is, the common ratio of the sequence is$$-1/3$$). We’ll call the nth number in this sequence \cl"red"{H(n)}. The fifth number in the sequence, \cl"red"{H(5)}, is$$-1/3(-2/3)=2/9$$. What is \cl"red"{H(6)}? In the table below, fill in the value of the expression$$-54 ⋅ (-1/3)^n$$for each value of n, and compare it to the nth number in the sequence, \cl"red"{H(n)}. n$$\cl"red"{H(n)}$$-54 ⋅ (-1/3)^n$$

We have seen that the geometric sequences $\cl"red"{G(n)}$ and $\cl"red"{H(n)}$ have exponential formulas: $G(n)=3 ⋅ 2^n$ and $$H(n)=-54 ⋅ (-1/3)^n$$. This is true of all geometric sequences.

In a geometric sequence with a common ratio of $r$, the $n$th number can be given by an exponential formula of the form $a ⋅ r^n$ for some number $a$.

Let’s look at the geometric sequence $J(n)$ which begins

$$6,18,54,...$$

Notice that $J(1)=6$, and the common ratio of this sequence is 3. Another way to say this is that the sequence is defined by:

$$J(n)=\{ \table 6, \text"if"\;n=1; 3J(n-1), \text"if"\;n>1$$

Because the common ratio is $3$, there must also be a formula for $J(n)$ which looks like $J(n)=a ⋅ 3^n$ for some number $a$. This formula tells us that $J(1)=a ⋅ 3^1 = 3a$.

 If $J(1)=3a$ and $J(1)=6$, then $3a=6$. What is the solution to the equation $3a=6$?
 What is a simple formula for $J(n)$?

## The Fibonacci sequence

In arithmetic and geometric sequences, each number depends on the number right before it. Now we’ll look at a sequence called the Fibonacci sequence, where each number depends on the two numbers before it. The Fibonacci sequence starts:

$$1,1,2,3,5,...$$

Each number after the first two is the sum of the two numbers before it. So the third number is $1+1=2$, the fourth number is $1+2=3$, and so on.

 The sixth number in this sequence is the sum of the fourth and fifth numbers, so it is $3+5=8$. What is the seventh number in the sequence?

Let $\cl"red"{F(n)}$ be the $n$th number in the Fibonacci sequence. Another way to give the definition of the Fibonacci sequence is to say that

$$\cl"red"{F(n)=\{ \table 1, \text"if"\;n=1\;\text"or"\;n=2; F(n-1)+F(n-2), \text"if"\;n>2}$$
 What is $\cl"red"{F(8)}$?
 The points $\cl"red"{(n,F(n))}$ are plotted on the grid to the left. Do they all lie on a straight line?
 Is the Fibonacci sequence an arithmetic sequence?

To see whether the Fibonacci sequence is a geometric sequence, we’ll look at the quantity $${F(n+1)}/{F(n)}$$ to see if it has a common ratio.

In the table below, find the value of $${F(n+1)}/{F(n)}$$ for each value of $n$, rounded to four decimal places. (Remember that the $≈$ symbol means “approximately equal to.”)