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AP Calculus AB & BC: Tutoring Solution17 chapters | 143 lessons

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Lesson Transcript

Instructor:
*Mimi Brestowski*

In this lesson, you will learn what an explicit formula is and how you can use it to identify terms in arithmetic and geometric sequences. The lesson also provides additional guidance on what arithmetic and geometric sequences are.

'Dad, what is an explicit formula?'

Like many other people in the world, math is my daughter's weakness, and it takes her several attempts, each with a changed delivery method, before she gets her 'Oh! I get it!' moment.

I explained to her that **explicit formula**, is a formula to find the *n*th term of a sequence. Her expression was blank, and I realized that in order to better present this to her, I would have to provide her with some background.

A sequence is a set of numbers that share a pattern. Each number in the sequence is a term. There are a total of 6 terms in this sequence on the screen. An infinite, or never ending, sequence is represent by three dots (...).

An **arithmetic sequence** has a pattern that adds or subtracts the same number to each term. Take the sequence of 0,2,4,6,8,10... Because we added two to each and every term, it is an arithmetic sequence.

A **geometric sequence** has a common ratio between terms. A sequence of 4, 12, 36, 108, 324... has a common ratio of 3.

Now that we know what the two different types of sequences are, arithmetic and geometric, we can dive into what exactly explicit means, what the formulas for the different sequences are, and how we can apply them to identify specific terms.

As mentioned, an explicit formula is a formula we can use to find the *n*th term of a sequence. In the easiest definition, explicit means exact or definite. The formula is explicit because as long as it's applied correctly, the *n*th term can be determined. Arithmetic and geometric sequences have different explicit formulas.

The explicit formula for an arithmetic sequence is *a* sub *n* = *a* sub 1 + *d*(*n*-1)

Don't panic! It'll make more sense once we break it down. The *a* sub *n* is made up of *a*, which represents a term, and the *n*, which represents the place of the term (first, second, third, etc.). The 1 we are subtracting from the *n* in the formula stands for the previous term. The next piece of the puzzle is *a* sub 1. This is your very first listed term of the sequence. Finally, *d* represent the difference between the terms or how many you add or subtract from term one to term two.

Now in three simple steps, let's use the previous arithmetic sequence of 0,2,4,6,8,10... to find out our explicit formula for it.

- State the arithmetic formula:
*a*sub*n*=*a*sub 1 +*d*(*n*-1) - What do you know? We know that the first term listed in the sequence is 0 so
*a*sub 1 = 0. We also know that we are adding 2 to each previous term so*d*= 2 - Plug what you know into the formula and solve the equation:

*a*sub*n*= 0 + 2(*n*- 1)*a*sub*n*= 0 + 2*n*- 2 (note: use the foil method 2 times*n*+ 2 times -1)*a*sub*n*= 2*n*- 2

And there you have it, the explicit formula for our arithmetic sequence is *a* sub *n* = 2*n* - 2!

You're probably thinking, 'what do I do with it now?' Well, you can now find any term within this infinite sequence. What if your homework requires you to find the 30th term of the sequence? No problem! Just plug in what you know into the explicit formula we just identified like this:

*a* sub *n* = 2*n* - 2 where *n* = 30 (since you are looking for the 30th term)

*a* sub 30 = 2(30) - 2

*a* sub 30 = 60 - 2 = 58

Your 30th term in the sequence is 58!

Let's move on to the explicit formula for a geometric sequence, which is *a* sub *n* = *a* sub 1 (*r*)^(*n*-1). Again our a sub 1 represents the first listed term of the geometric sequence, *n* represent the place of the term, and *r* represents our common ratio.

To keep it simple, we will use the previously mentioned geometric sequence of 4, 12, 36, 108, 324,â€¦ We have also previously determined that our common ratio was 3. Recall that we got this ratio by taking each subsequent term and dividing it by its predecessor. Now just plug in what you know to get the explicit formula for this particular geometric sequence.

- State the geometric formula:
*a*sub*n*=*a*sub 1(*r*)^(*n*-1) - What do you know? We know that the first term listed in the sequence is 4 so
*a*sub 1 = 4. We also know that we are multiplying each term by 3 so*r*= 3. - Plug what you know into the formula and solve the equation:

*a*sub*n*=*a*sub 1(*r*)^(*n*-1)*a*sub*n*= 4(3)^(*n*-1)

Now that we have our explicit formula for the geometric sequence, we can find any term within that sequence. Suppose you want to find the 10th term of the sequence. Plug in what you know into the formula like this:

*a* sub *n* = 4(3)^(*n*-1) (where n = 10 since you are looking for the 10th term)

*a* sub 10 = 4(3)^(10-1)=9

*a* sub 10 = 4(3)^9 = 4(19,683) (note: don't forget your exponents get solved before multiplication)

*a* sub 10 = 78,732

Your 10th term in the sequence is 78,732!

You have just learned that a **sequence** is a set of terms with a clear pattern. It can be arithmetic or geometric in nature. You've also learned that an **explicit formula** is used to find the *n*th term of **arithmetic sequence** or **geometric sequence** through using the following formulas:

The **arithmetic explicit formula** is *a* sub *n* = *a* sub 1 + *d*(*n*-1)

The **geometric explicit formula** is *a* sub *n* = *a* sub 1 (*r*)^(*n*-1)

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AP Calculus AB & BC: Tutoring Solution17 chapters | 143 lessons

- What is a Mathematical Sequence? 5:37
- How to Find and Classify an Arithmetic Sequence 9:09
- Finding and Classifying Geometric Sequences 9:17
- Summation Notation and Mathematical Series 6:01
- How to Calculate an Arithmetic Series 5:45
- How to Calculate a Geometric Series 9:15
- Arithmetic and Geometric Series: Practice Problems 10:59
- Explicit Formula & Sequences: Definition & Examples 6:51
- Go to Sequences and Series in AP Calculus: Tutoring Solution

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