College Algebra
Answer/Discussion to Practice Problems
Tutorial 54D: Geometric Sequences and Series
Answer/Discussion
to 1a

Since a geometric sequence is a sequence, you find
the terms exactly the same way that you do a sequence. n is our term number and we
plug the term number into the function to find the value of the term.
If you need a review on sequences, feel free to go to Tutorial 54A: Sequences.
Lets see what we get for
our first five terms:

What
would be the common ratio for this problem?
If you said 5 you are correct!
Note that you would have to multiply (10)(5) = 50, (50)(5) = 250,
(250)(5) = 1250, and (1250)(5) = 6250. It has to be
consistent throughout the sequence.
Also note that the base that is being raised to a power is 5. 
Answer/Discussion
to 1b

Since a geometric sequence is a sequence, you find
the terms exactly the same way that you do a sequence. n is our term number and we
plug the term number into the function to find the value of the term.
If you need a review on sequences, feel free to go to Tutorial 54A: Sequences.
Lets see what we get for
our first five terms:

What
would be the common ratio for this problem?
If you said 1/4 you are correct!
Note that you would have to multiply by 1/4 each time you go from one
term to the next: (4)(1/4) = 1, (1)(1/4) = 1/4, (1/4)(1/4) =
1/16, and (1/16)(1/4)=1/64. It has to be consistent
throughout the sequence.
Also note that the base that is being raised to a power is 1/4. 
Answer/Discussion
to 2a
125, 25, 5, 1, ... 
We will use the nth
term formula for a geometric sequence, to help us with
this problem.
Basically we need to find two things: the first term of the
sequence, and the
common ratio, r.
What
is , the first term?
If you said 125, give yourself a high five. The first term of
this sequence is 125.
What
is r, the common ratio?
If you said 1/5, you are right!!
Note that you would have to multiply 1/5 each time you go from one term
to the next: (125)(1/5) = 25, (25)(1/5) = 5, and (5)(1/5) =
1. It has to be consistent throughout the sequence.
Putting in 125 for and 1/5 for r we get:

Answer/Discussion
to 2b
4, 12, 36, 108, ...

We will use the nth
term formula for a geometric sequence, to help us with
this problem.
Basically we need to find two things: the first term of the
sequence, and the
common ratio, r.
What
is , the first term?
If you said 4, give yourself a high five. The first term of this
sequence is 4.
What
is r, the common ratio?
If you said 3, you are right!!
Note that you would have to multiply 3 each time you go from one term
to the next: (4)(3) = 12, (12)(3) = 36, and (36)(3) =
108. It has to be consistent throughout the sequence.
Putting in 4 for and 3 for r we get:

Answer/Discussion
to 3a
2 + 14 + 98 + 686 + 4802 + 33614 + 235298

We will use the formula for the sum of the first n terms of geometric
sequence, , to
help us with this problem.
Basically we need to find three things: the first term of the sequence,
the common ratio, and how many terms of the sequence we are adding in
the series.
What
is , the first term?
If you said 2 you are right!
What
is r, the common
ratio?
If you said 7, give yourself a pat on the back. Note that you would
have to multiply 2 each time you go from one term to the next: (2)(7)
= 14, (14)(7) = 98, (98)(7) = 686, (686)(7) = 4802, (4802)(7) = 33614,
and (33614)(7) = 235298. It has to be consistent throughout
the sequence.
How
many terms are we summing up?
If you said 7, you are correct.
Putting in 2 for the first
term, 7 for the common ratio, and 7 for n, we get:

Answer/Discussion
to 3b

We will use the formula for the sum of the first n terms of geometric
sequence, , to
help us with this problem.
Basically we need to find three things: the first term of the sequence,
the common ratio, and how many terms of the sequence we are adding in
the series.
What
is , the first term?
If you said 12.25 you are right!
Since this summation starts at 2, you need to plug in 2 into the given
formula:
What
is r, the common
ratio?
If you said 3.5, give yourself a pat on the back. Note that 3.5
is the number that is being raised to the exponent. So each time
the number goes up on the exponent, in essence you are multiplying it
by 3.5.
How
many terms are we summing up?
If you said 9, you are correct. If you start at 2 and go all the
way to 10, there will be 9 terms.
Putting in 12.25 for the
first
term, 3.5 for the common ratio, and 9 for n, we get:

Answer/Discussion
to 4a

We will use the formula for the sum of infinite
geometric sequence, , to help us with this problem.
Basically we need to find two things: the first term of the sequence
and the common ratio.
What
is the first term, ?
If you said 1 you are right!
Since this summation starts at 0, you need to plug in 0 into the given
formula:
What
is the common ratio, r?
If you said .1, give yourself a pat on the back. Note that .1 is the
number that is being raised to the exponent. So each time the
number goes up on the exponent, in essence you are multiplying it by
.1.
Putting in 1 for the
first term and .1 for the common ratio we get:

Answer/Discussion
to 4b

We will use the formula for the sum of infinite
geometric sequence, , to help us with this problem.
Basically we need to find two things: the first term of the sequence
and the common ratio.
What
is the first term, ?
If you said 1 you are right!
What
is the common ratio, r?
If you said 3/2, give yourself a pat on the back. Note that you would
have to multiply 3/2 each time you go from one term to the next:
(1)(3/2)
= 3/2, (3/2)(3/2) = 9/4, (9/4)(3/2) = 27/8. It has to
be consistent
throughout the sequence.
Since the geometric ratio
is 3/2 and ,
there
is no sum.

Last revised on May 16, 2011 by Kim Seward.
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