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# 7.2: Explicit Formulas

Difficulty Level: At Grade Created by: CK-12
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Rachel and Elaina have started a website where they debate the best color of hair dye. The show is really popular and the visitors to their website are increasing very rapidly. They figure that memberships are increasing by about 500 people every three days.

At this rate, how many members will they have on the 48th day? How many days will it be before they reach 25,000 members?

### Watch This

Arithmetic Sequences:

- Brightstorm: Arithmetic Sequences

Geometric Sequences:

http://youtu.be/cDRkMVTM0SE - Brightstorm: Geometric Sequences]

### Guidance

When we represent a sequence with a formula that lets us find any term in the sequence without knowing any other terms, we are representing the sequence explicitly.

Given a recursive definition of an arithmetic or geometric sequence, you can always find an explicit formula, or an equation to represent the nth term of the sequence. Consider for example the sequence of odd numbers we started with: 1,3,5,7,...

We can find an explicit formula for the nth term of the sequence if we analyze a few terms:

a1 = 1
a2 = a1 + 2 = 1 + 2 = 3
a3 = a2 + 2 = 1 + 2 + 2 = 5
a4 = a3 + 2 = 1 + 2 + 2 + 2 = 7
a5 = a4 + 2 = 1 + 2 + 2 + 2 + 2 = 9
a6 = a5 + 2 = 1 + 2 + 2 + 2 + 2 + 2 = 11

Note that every term is made up of a 1, and a set of 2’s. How many 2’s are in each term?

a1 = 1
a2 = 1 + 2 = 3
a3 = 1 + 2 × 2 = 5
a4 = 1 + 3 × 2 = 7
a5 = 1 + 4 × 2 = 9
a6 = 1 + 5 × 2 = 11

The nth term has (n - 1) 2's. For example, a99 = 1 + 98 × 2 = 197 . We can therefore represent the sequence as an = 1 + 2(n - 1). We can simplify this expression:

an = 1 + 2(n - 1)
an = 1 + 2n - 2
an = 2n - 1

In general, we can represent an arithmetic sequence in this way, as long as we know the first term and the common difference, d. Notice that in the previous example, the first term was 1, and the common difference, d, was 2. The nth term is therefore the first term, plus d(n - 1):

an =a1 + d(n - 1)

You can use this general equation to find an explicit formula for any term in an arithmetic sequence.

#### Example A

Find an explicit formula for the nth term of the sequence 3, 7, 11, 15... and use the equation to find the 50th term in the sequence.

Solution

an = 4n - 1 , and a50 = 199

The first term of the sequence is 3, and the common difference is 4.

an = a1 + d(n - 1)
an = 3 + 4(n - 1)
an = 3 + 4n - 4
an = 4n - 1
a50 = 4(50) - 1 = 200 - 1 = 199

We can also find an explicit formula for a geometric sequence. Consider again the sequence in example 2a:

t2 = 2t1 = 2 × 3 = 6
t1 = 3 \begin{align*}\rightarrow\end{align*} t3 = 2t2 = 2 × 6 = 12
tn = 2 × tn-1 t4 = 2t3 = 2 × 12 = 24
t5 = 2t4 = 2 × 24 = 48

Notice that every term is the first term, multiplied by a power of 2. This is because 2 is the common ratio for the sequence.

t1 = 3
t2 = 2 × 3 = 6
t3 = 2 × 2 × 6 = 22 × 6 = 12
t4 = 2 × 2 × 2 × 6 = 23 × 6 = 24
t5 = 2 × 2 × 2 × 2 × 6 = 24 × 6 = 48

The power of 2 in the nth term is (n-1). Therefore the nth term in this sequence can be defined as: tn = 3(2n - 1). In general, we can define the nth term of a geometric sequence in terms of its first term and its common ratio, r:

tn = t1(rn-1)

You can use this general equation to find an explicit formula for any term in a geometric sequence.

#### Example B

Find an explicit formula for the nth term of the sequence 5, 15, 45, 135... and use the equation to find the 10th term in the sequence.

Solution

an = 5 × 3n - 1, and a10 = 98,415

The first term in the sequence is 5, and r = 3.
an = a1 × rn - 1
an = 5 × 3n - 1
a10 = 5 × 310 - 1
a10 = 5 × 39 = 5 × 19,683 = 98,415

Again, it is always possible to write an explicit formula for terms of an arithmetic or geometric sequence. However, you can also write an explicit formula for other sequences, as long as you can identify a pattern. To do this, you must remember that a sequence is a function, which means there is a relationship between the input and the output. That is, you must identify a pattern between the term and its index, or the term’s “place” in the sequence.

#### Example C

Write an explicit formula for the nth term of the sequence 1, (1/2), (1/3), (1/4)...

Solution

an = (1/n)

Initially you may see a pattern in the fractions, but you may also wonder about the first term. If you write 1 as (1/1), then it should become clear that the nth term is (1/n).

Concept question wrap-up "They figure that memberships are increasing by about 500 people every three days. At this rate, how many members will they have on the 48th day? How many days will it be before they reach 25,000 members?"

This is actually a fairly simple arithmetic sequence: each day there are 500/3 more members, on average. Use the formula for arithmetic sequences from the first example above.

### Vocabulary

An explicit formula for a sequence allows you to find the value of any term in the sequence.

The natural numbers are a subset of the integers: {1,2,3,4,5....}

A recursive formula for a sequence allows you to find the value of the nth term in the sequence if you know the value of the(n-1)th term in the sequence.

A sequence is an ordered list of objects.

### Guided Practice

Questions

1) Write an explicit formula for the sequence: 2, 9, 16... and use the formula to find the value of the 20th term.

2) Write an explicit formula for the sequence: 5, 10, 20... and use the formula to find the value of the 9th term.

3) Write an explicit formula for the sequence: (1/2), (1/4), (1/8) and use the formula to find the value of the 7th term.

4) Identify all sequences in the previous three problems that are geometric. What is the common ratio in each sequence?

5) The membership of an online dating service increases at an average rate of 8% per year. In the first year, there are 500 members.

a. How many members are there in the second year?
b. How many members are there in the eighteenth year?

Solutions

1) For the sequence: 2, 9, 16...

an=7n5\begin{align*}a_n = 7n - 5\end{align*}
a20=7(20)5\begin{align*}\therefore a_{20} = 7(20) - 5\end{align*}
a20=135\begin{align*}a_{20} = 135\end{align*}

2) For the sequence: 5, 10, 20...

an=52n1\begin{align*}a_n = 5 \cdot 2^{n-1}\end{align*}
a9=528\begin{align*}\therefore a_9 = 5 \cdot 2^8\end{align*}
a9=5256\begin{align*}a_9 = 5 \cdot 256\end{align*}
a9=1280\begin{align*}a_9 = 1280\end{align*}

3) For the sequence: (1/2), (1/4), (1/8)...

an=12n\begin{align*}a_n = \frac{1}{2^n}\end{align*}
a7=127\begin{align*}\therefore a_7 = \frac{1}{2^7}\end{align*}
a7=1128\begin{align*}a_7 = \frac{1}{128}\end{align*}

4) The sequence in question 1 is arithmetic.

The sequence is question 2 is geometric, and has r = 2.

The sequence in question 3 is geometric and has r = 1/2.

5) a. 540 members

b. Approximately 1,998 members

### Practice

Name the sequence as arithmetic, geometric, or neither.

1. 21,6,18,3,20,2\begin{align*}-21, -6, 18, -3, 20, -2\end{align*}
2. 0,15,25,35,45,1\begin{align*}0,\frac{-1}{5}, \frac{-2}{5}, \frac{-3}{5}, \frac{-4}{5}, -1\end{align*}
3. 1,3,9,27,81,243\begin{align*}1, 3, 9, 27, 81, 243\end{align*}
4. 2,9,2,1,18,2\begin{align*}2, 9, -2, 1, 18, 2\end{align*}

Write the first 5 terms of the arithmetic sequence(explicit).

1. an=89(n1)\begin{align*}a_n = -8 -9(n-1)\end{align*}
2. an=623(n1)\begin{align*}a_n = 6-\frac{2}{3}(n-1)\end{align*}
3. an=8+13(n1)\begin{align*}a_n = 8 + \frac{1}{3}(n-1)\end{align*}

Solve the following:

1. What are the first five terms of the sequence? an=an1103;a1=6\begin{align*}a_n = a_{n-1} - \frac{10}{3}; a_1 = -6\end{align*}
2. Given the sequence, write a recursive function to generate it: 2,4,10,16,22,28\begin{align*}2, -4, -10, -16, -22, -28\end{align*}
3. Write the equation of an\begin{align*}a_n\end{align*} without using recursion: an=an132;a1=10\begin{align*}a_n = a_{n-1} - \frac{3}{2}; a_1 = 10\end{align*}
4. Write as a recursion: an=653(n1)\begin{align*}a_n = 6 - \frac{5}{3}(n-1)\end{align*}
5. Write the equation of an\begin{align*}a_n\end{align*} without using recursion: an=an1+8;a1=3\begin{align*}a_n = a_{n-1}+8; a_1 = 3\end{align*}
6. What are the first five terms of the sequence? an=an11;a1=5\begin{align*}a_n = a_{n-1} - 1; a_1 = -5\end{align*}

Write the formula for the explicit(nth)\begin{align*}(n_{th})\end{align*}term of the arithmetic sequence.

1. 7,133,53,1,113,193\begin{align*}-7, \frac{-13}{3}, \frac{-5}{3}, 1, \frac{11}{3}, \frac{19}{3}\end{align*}
2. 6,4,14,24,34,44\begin{align*}6, -4, -14, -24, -34, -44\end{align*}
3. 9,16,23,30,37,44\begin{align*}9, 16, 23, 30, 37, 44\end{align*}
4. In a particular arithmetic sequence, the second term is 4 and the fifth term is 13. Write an explicit formula for this sequence.

Write the first 5 terms of the geometric sequence

1. an=5(3)(n1)\begin{align*}a_n = 5(-3)^{(n-1)}\end{align*}
2. an=6(103(n1))\begin{align*}a_n = -6(\frac{-10}{3}^{(n-1)})\end{align*}

Write the formula for the nth\begin{align*}n_{th}\end{align*} term of the geometric sequence

1. 8,16,32,64,128,256\begin{align*}-8, 16, -32, 64, -128, 256\end{align*}
2. 9,272,814,2438,72916,218732\begin{align*}9, \frac{27}{2}, \frac{81}{4}, \frac{243}{8}, \frac{729}{16}, \frac{2187}{32}\end{align*}

Convert the explicit and rewrite as recursion:

1. an=9(43)(n1)\begin{align*}a_n = 9(\frac{-4}{3})^{(n-1)}\end{align*}
2. an=6(4)(n1)\begin{align*}a_n = -6(-4)^{(n-1)}\end{align*}
3. an=5(5)(n1)\begin{align*}a_n = -5(5)^{(n-1)}\end{align*}

### Vocabulary Language: English

arithmetic sequence

arithmetic sequence

An arithmetic sequence has a common difference between each two consecutive terms. Arithmetic sequences are also known are arithmetic progressions.
common difference

common difference

Every arithmetic sequence has a common or constant difference between consecutive terms. For example: In the sequence 5, 8, 11, 14..., the common difference is "3".
common ratio

common ratio

Every geometric sequence has a common ratio, or a constant ratio between consecutive terms. For example in the sequence 2, 6, 18, 54..., the common ratio is 3.
Explicit

Explicit

Explicit formulas define each term in a sequence directly, allowing one to calculate any term in the sequence without knowing the value of the previous terms.
Explicit formula

Explicit formula

Explicit formulas define each term in a sequence directly, allowing one to calculate any term in the sequence without knowing the value of the previous terms.
geometric sequence

geometric sequence

A geometric sequence is a sequence with a constant ratio between successive terms. Geometric sequences are also known as geometric progressions.
index

index

The index of a term in a sequence is the term’s “place” in the sequence.
Natural Numbers

Natural Numbers

The natural numbers are the counting numbers and consist of all positive, whole numbers. The natural numbers are the numbers in the list 1, 2, 3... and are often referred to as positive integers.
recursive

recursive

The recursive formula for a sequence allows you to find the value of the nth term in the sequence if you know the value of the (n-1)th term in the sequence.
recursive formula

recursive formula

The recursive formula for a sequence allows you to find the value of the nth term in the sequence if you know the value of the (n-1)th term in the sequence.
sequence

sequence

A sequence is an ordered list of numbers or objects.

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