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(Video) Explicit Formulas

by fethermath

The video above by fethermath gives a more general description of explicit formula.

$\large{\text{Summary}}$

An explicit formula helps you calculate the $n^{\text{th}}$ term of a sequence explicitly, or directly, by substituting a value for $n$.

Recall that $n$ is just the term number we want to find. It's the position of the term we want in the sequence. So if you want to find the third term in a sequence, you let $n=3$.

Suppose we have the following sequence: $$\begin{array}{c|c|c} \text{Term number}&n&1&2&3&4&5&6\\ \hline \text{Term}&t_n&4&8&12&16&20&24 \end{array}$$

So how do we express this formula? We want to be able to express any term in the sequence, $t_n$, using $n$.

Start by looking at the first term. How do we get from the term number, $1$, to the term value $4$? We could multiply $1$ by $4$, or we can add $3$ to $1$.

Now look at the second term. How do we get from the term number, $2$, to the term value $8$? We could multiply $2$ by $4$, or we can add $6$ to $2$.

Notice a consistent pattern already. We can multiply the term number by $4$ to get the term value. Therefore, we can write our explicit formula for the $n^{\text{th}}$ term as: $$t_n=4n$$

We can use this to find specific terms. Suppose we want to know the $8^{\text{th}}$ term. Just plug in $n=8$. $$\begin{align} t_n&=4n\\ t_8&=4(8)\\ t_8&=32 \end{align}$$

Let's do another practice problem.

We need to plug in $1, 2,$ and $3$ for $n$. This will give us each of the first three terms of our sequence. $$\begin{align}a_n&=3^n\\ \\ a_1&=3^1\\ a_1&=3\\ \\ a_2&=3^2\\ a_2&=9\\ \\ a_3&=3^3\\ a_3&=27 \end{align}$$

The first three terms are $3,9,27$.

(Video) Writing Explicit Formula of a Sequence

by Brian McLogan

Here's a quick but great video by Brian McLogan on how to figure out the explicit formula for a sequence.

$\large{\text{Summary}}$

An explicit formula calculates the $n^{\text{th}}$ term by substituting a value in for $n$.

For example, suppose we have the sequence: $$2,4,6,8,\cdots$$ The explicit formula for the $n^{\text{th}}$ term of the sequence is: $$a_n=2n$$

If you want to find the fourth term, you plug in $n=4$. $$a_4=2(4)=8$$

How Though?

How do you figure out the equation in the first place? Well, there's not always an easy way. Mostly, you have to test some theories until you find the one that works. This is demonstrated in the video above.

Suppose we want to find the explicit formula for the sequence: $$3,7,11,15,19,\cdots$$

A good strategy is to find a formula that works for the first few terms and apply it to the next few terms. For the most part (not always, though), if it works for the a few terms, it works for the sequence.

We start by asking, how can we get from the position number, $n=1$, to the term number, $3$. Well, we could multiply $1$ by $3$. If this is the pattern, then: $$a_1=3(1)\quad\rightarrow\quad a_n=3n$$

Let's now test it on the second term, where $n=2$. We know it should equal $7$. $$\begin{align} a_n&=3n\\ a_2&=3(2)\\ a_2&=6 \end{align}$$ This doesn't work. So we can scrap that formula.

Back to square one. How else can we get from the term position, $1$, to the term $3$. We can add $2$ to $1$. This means: $$a_1=1+2\quad\rightarrow\quad a_n=n+2$$

Again, we test it on the second term which we know is $7$. $$\begin{align} a_n&=n+2\\ a_2&=2+2\\ a_2&=4 \end{align}$$ This also doesn't work.

Ugh, fine. What now? We can try combining multiplication and addition. If we multiply $1$ by $2$ and then add $1$, we can get to $3$. $$a_1=2(1)+1\quad\rightarrow\quad a_n=2n+1$$

You know the drill. Test it. $$\begin{align} a_n&=2n+1\\ a_2&=2(2)+1\\ a_2&=5 \end{align}$$ This doesn't work either.

So now what? Well, we've tried multiplying by $2$ and multiplying by $3$. Let's try multiplying by $4$. If we multiply $1$ by $4$ we need to then subtract $1$ to get to $3$. $$a_1=4(1)-1\quad\rightarrow\quad a_n=4n-1$$

Alright, let's test it. $$\begin{align} a_n&=4n-1\\ a_2&=4(2)-1\\ a_2&=7 \end{align}$$ This worked! Therefore, we can infer that the explicit formula for the sequence is: $$a_n=4n-1$$

If you check this formula with the other terms of the sequence, you will see that it works!

You can also use this method for arithmetic sequences!

Practice Problem

Since these are the positive odd numbers, you can find that the explicit formula for the sequence is: $$a_n=2n-1$$ Once you get this, you can find the $19^{\text{th}}$ term by plugging in $n=19$. $$\begin{align} a_n&=2n-1\\ a_{19}&=2(19)-1\\ a_{19}&=38-1\\ a_{19}&=37 \end{align}$$

Maximum Depth Exceeded: Fun with Sequences and Python, Part 3

To read part $2$ of this story, go here.

If we are asked to find an explicit formula for a sequence, it helps to start with a recurrence formula, as we can use this to find an explicit formula. An explicit formula for a sequence is one for which $a_n$ equals something that does not depend on any other elements in the sequence.

For example, let's work with the recurrence formula

$$\begin{align} a_n &= a_{n - 1} - 1 \\ a_0 &= 7. \end{align}$$

We can plug this formula into Python and get the following values in the sequence:

What we've done with Python is listed out the first $21$ terms of the sequence, in order. Each term is one less than the previous term. This means that after $n$ terms, our value is $n$ less than the starting term, meaning that our sequence has an explicit form:

$$a_n = a_0-n=7 - n$$

because we subtract $1$ off every time we move to the next term, and we start the sequence from $a_0=7.$ This formula is called "explicit" because if we are trying to compute $a_n,$ there aren't any other appearances of the sequence $a_0, a_1, a_2,...$ in the formula, while our recurrence formula

$$a_n = a_{n - 1} - 1$$

had $a_{n - 1}$ in it, so it's not explicit.

To go to part $4$ of this story, which covers geometric sequences, go here.

Explicit Formulas for Sequences

An explicit formula allows you to you directly calculate the $n^{th}$ term in a sequence. It makes it easy to calculate whatever term you want. All you have to know is know the term's position! For example, say you wanted to know the $45^{th}$ term in a sequence, and you don't have the time to try to figure out all the terms before it. If you have an explicit formula, you can directly calculate the $45^{th}$ term!

Explicit formulas take the form $$a_n=f(n)$$

where $n$ represents the position number of the term. Basically, an explicit formula represents the $n^{th}$ term ($a_n$) as a function of its position number ($f(n)$).

Let's do another example. Say you are given the explicit formula $a_n=2n^2+5,$ and you want to find the $11^{th}$ term. All you have to do is plug in $11$ for $n.$

$$\begin{align} a_n&=2n^2+5\\ a_{11}&=2(11)^2+5\\ a_{11}&=2(121)+5\\ a_{11}&=242+5\\ a_{11}&=247 \end{align}$$