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Understanding Geometric Sequences: Explicit and Recursive Formulas, Lecture notes of Analytical Geometry and Calculus

University of Arkansas Community College - MorriltonAnalytical Geometry and Calculus

The concept of geometric sequences and demonstrates how to find the explicit and recursive formulas for them. Geometric sequences are a type of sequence where each term is obtained by multiplying the previous term by a constant. Examples and formulas for both explicit and recursive formulas, as well as instructions for translating between the two.

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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Geometric Sequences Explicit and Recursive Formulas
Recall that a geometric sequence has a pattern of multiplying by the same value over and
over again. We can model that type of sequence in two ways. One is called an explicit
formula and the other a recursive formula. First lets look at explicit formulas.
Explicit formula of a Geometric Sequences
The explicit form of a sequence is used to find the general term, or "nth" term, by plugging in
the number of term we want to know. The explicit form of a geometric sequence is
(
((
(
)
))
)
1
1
−
−−
−
=
==
=
n
n
rtt
where
n
t
is the general term,
1
t
is the first term of the sequence,
r
is the
common ratio, and
n
is the number of term to plug in.
For example:
In the equation
(
((
(
)
))
)
1
26
−
−−
−
=
==
=
n
n
t
6 is the first term of the sequence and 2 is
the common ratio. We could write out the first several terms of this sequence and get
6, 12, 24, 48, ... Suppose we want to know what the 10th term of this sequence is without
having to go through finding all of the middle terms. We can first plug in 10 for
n
in the
formula. Then use order of operations to evaluate for the tenth term.
(
((
(
)
))
)
(
((
( )
))
)
(
((
( )
))
)
(
((
( )
))
)
3072
5126
26
26
26
10
10
9
10
110
10
1
=
==
=
=
==
=
=
==
=
=
==
=
=
==
=
−
−−
−
−
−−
−
t
t
t
t
t
n
n
So the tenth term of this sequence is 3072.
Likewise we can write an explicit formula for a geometric sequence by plugging in the first
term and common ratio into
(
((
(
)
))
)
1
1
−
−−
−
=
==
=
n
n
rtt
.
For example:
In the sequence 2, 6, 18, 54, ...
To write the explicit formula for this sequence we need to know the first term and the
common ratio. We can see that the first term is 2. To find the common ratio divide any
successive pair of terms such that the second in the pair is always the numerator and the
first in the pair is the denominator.
3
2
6
6&2
=
==
==
==
=r 3
6
18
18&6
=
==
==
==
=r 3
18
54
54&18
=
==
==
==
=r
Now we know that the common ratio is 3.
Using
2
1
=
==
=
t and
3
=
==
=
r
we get
(
((
(
)
))
)
1
32
−
−−
−
=
==
=
n
n
t
pf2

Partial preview of the text

Download Understanding Geometric Sequences: Explicit and Recursive Formulas and more Lecture notes Analytical Geometry and Calculus in PDF only on Docsity!

Geometric Sequences Explicit and Recursive Formulas

Recall that a geometric sequence has a pattern of multiplying by the same value over and

over again. We can model that type of sequence in two ways. One is called an explicit

formula and the other a recursive formula. First lets look at explicit formulas.

Explicit formula of a Geometric Sequences

The explicit form of a sequence is used to find the general term, or "nth" term, by plugging in

the number of term we want to know. The explicit form of a geometric sequence is

1

1

−−−−

n

n

t t r where

n

t is the general term,

1

t is the first term of the sequence, r is the

common ratio, and n is the number of term to plug in.

For example: In the equation (((( ))))

1

−−−−

n

n

t 6 is the first term of the sequence and 2 is

the common ratio. We could write out the first several terms of this sequence and get

6, 12, 24, 48, ... Suppose we want to know what the 10th term of this sequence is without

having to go through finding all of the middle terms. We can first plug in 10 for n in the

formula. Then use order of operations to evaluate for the tenth term.

10

10

9

10

10 1

10

1

−− −−

−−−−

t

t

t

t

t

n

n

So the tenth term of this sequence is 3072.

Likewise we can write an explicit formula for a geometric sequence by plugging in the first

term and common ratio into (((( ))))

1

1

−−−−

n

n

t t r.

For example: In the sequence 2, 6, 18, 54, ...

To write the explicit formula for this sequence we need to know the first term and the

common ratio. We can see that the first term is 2. To find the common ratio divide any

successive pair of terms such that the second in the pair is always the numerator and the

first in the pair is the denominator.

r = 3

r ==== ==== 3

r ==== == ==

Now we know that the common ratio is 3.

Using 2

1

t ==== and r ==== 3 we get (((( ))))

1

−−−−

n

n

t

Recursive formula of a Geometric Sequence

The recursive form of a sequence is slightly different. For a recursive equation we much be

given one term (usually the first term) and how to use that term to get to the next term. Often

we see this written as NEXT/NOW form. Let's look at our previous example

2, 6, 18, 54, ...

Suppose I tell you that the first term is 2. How can I get from 2 to 6 or from 6 to 18 or from 18

to 54? Clearly we can multiply by 3 each time. So one way to show that would be to say

NEXT ==== NOW 3 since each term is three times the previous term.

To write that recursively we plug it into the following form for a recursive geometric

sequence. ____; (((( )))) for 1

1 1

−−−−

t t t r n

n n

Notice that NEXT is replaced with

n

t and

NOW is replaced with

n −−−− 1

t. Plug in the correct values for first term and common ratio just

like we did for the explicit formula.

2 ; 3 for 1

____; for 1

1 1

1 1

−− −−

−−−−

t t t n

t t t r n

n n

n n

Translating between Explicit and Recursive formulas

We can clearly se that both the explicit and recursive forms use the same two key pieces of

information about the sequence, the first term and the common ratio. That being the case it

is fairly easy to translate between the two. Remember the explicit form is (((( ))))

1

1

−−−−

n

n

t t r and

the recursive form is ____; (((( )))) for 1

1 1

−−−−

t t t r n

n n

. We simply need to know where to

look to find the first term and the common ratio to change forms.

Example: Write the following explicit geometric sequence as a recursive geometric

sequence.

1

− −−

−

n

n

t

We can now identify

1

t as 18 and r as 1.5. So we can plug them into the recursive form

and get 18 ; (((( 1. 5 )))) for 1

1 1

−−−−

t t t n

n n

.

Example: Write the following recursive geometric sequence as a explicit geometric

sequence.

for 1

1 1

−−−−

t t t n

n n

We can now identify

1

t as 30 and r as 1/3. So we can plug them into the explicit form

and get

1

−−−−

n

n

t