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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.
In the sequence of numbers
each number is 2 larger than the previous one.
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.
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 $n$.
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.
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:
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.
We will call the $n$th number in this sequence $\cl"red"{B(n)}$. For example, $B(2)=4$ because 4 is the 2nd number in the sequence.
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 $n$.
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 $n$th 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
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:
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 $C(1)=3(1)+b=3+b$.
Now we’ll look at the sequence
where each number is 2 times the previous number. Call the $n$th number in this sequence $\cl"red"{G(n)}$.
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.
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 $n$.
Compute the quantities $${G(n+1)}/{G(n)}$$ in the table below.
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:
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 $n$th 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 $n$th number in the sequence, $\cl"red"{H(n)}$.
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
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:
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 $J(1)=a ⋅ 3^1 = 3a$.
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:
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.
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
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.”)
Notice that the numbers $${F(n+1)}/{F(n)}$$ are not equal, but they are close to each other. After the first few, they are all a little bit more than 1.6. This means that the Fibonacci sequence is not a geometric sequence, but it is almost a geometric sequence.
Specifically, when $n$ is a large number, $\cl"red"{F(n)}$ is very close to $$1/√5 ⋅ r^n$$, where $$r={1+√5}/{2} ≈ 1.6180$$. For example, the 20th number in the Fibonacci sequence is
and $$1/√5 ⋅ r^20$$ is approximately
Let’s look at a sequence $T(n)$ which begins
Here $T(1)=1$, $T(2)=1+2=3$, $T(3)=1+2+3=6$, and so on: that is, $T(n)$ is the number you get by adding up the first $n$ positive integers. This is pictured on the grid to the left, with a slider for $n$.
When $n$ is large, you need to add up a lot of numbers to find $T(n)$! Let’s look for an easier formula.
Click to see a picture of two copies of $T(n)$. One of them has been turned upside-down so they fit together into a rectangle, whose area must be equal to $T(n)+T(n)$.
For each value of $n$ in the table below, slide the slider to $n$ in order to find the width, height, and area of this rectangle.
As you can see, two copies of $T(n)$ fit together to form a rectangle. This rectangle has width $n$ and height $n+1$, making its area $n(n+1)$. This means that:
So we’ve found a formula for $T(n)$: $$T(n)={n(n+1)}/2$$.
Using this formula, what is $T(100)$?