Download Precalculus Honors Sequences and Series September 12, 2005 Mr

Precalculus Honors
Mr. DeSalvo
1.4
Sequences and Series
September 12, 2005
GEOMETRIC SEQUENCES AND SERIES
Imagine dropping a ball from a height of 4 feet such that it bounces
straight up and down reaching a height that is exactly half of its previous height.
Finish filling in the partial table below showing the maximum height achieved by
the ball after the nth bounce.
Bounce, n
Height (in feet)
0
4
1
2
2
3
4
5
If you catch the ball when it reaches the top after 5th bounce, how far will the ball
have traveled? To answer this question, we must add the distances the ball drops
and then rises, as follows: down 4 feet, up 2 feet, down 2 feet, up one foot, and so
on. This leads to the following sequence of “down-up” distances:
6, 3,
3 3 3
,
,
2 4 8
Notice that each new term in the sequence can be obtained by multiplying the
1
previous term by a factor of . This is an example of a geometric sequence.
2
DEFINITION OF A GEOMETRIC SEQUENCE
A sequence is said to be geometric if each term, after the first, is obtained by
multiplying the preceding term by a common value.
Now back to the original question: how far has the ball traveled during the five
bounces? To answer this question, we add the geometric sequence as follows:
6 + 3+
3 3 3 48 + 24 + 12 + 6 + 3 93
5
+ + =
=
= 11
2 4 8
8
8
8
Thus, if we catch the ball after it reaches the top of the fifth bounce, it will have
5
traveled a total of 11 feet.
8
EXERCISES The first three terms of a geometric sequence are given, find the
initial term a1 and the common ratio r. The first one is done for you.
1. 3, 6, 12, . . .
2. 2, -4, 8, . . .
3. 100, 10, 1, …
8
4. -6, -4, − , . . .
3
a1 = 3; r = 2
5. Find the next three terms of each geometric sequence in 1 – 4 above.
Note: the common ratio r is found by dividing any term into the next one.
GENERAL TERM OF A GEOMETRIC SEQUENCE
The nth term of a geometric sequence is
an = a1r n−1
Where a1 is the first term and r is the common ratio.
Verify this formula by using it to produce a6 for each sequence in exercises 1 - 4 above.
27
, . . . and then
2
find the seventh term. (Hint: first find r and then use the above formula to find the
nth term.)
EXAMPLE 1 Find the nth term of the geometric sequence 6, 9,
Solution:
r = 9/6 = 3/2, so an = 6(3/2)n-1. Thus a7 = 6(3/2)7-1 = 6(3/2)6 =
2187
.
32
EXAMPLE 2 A geometric sequence consisting of positive numbers has a1 = 18
32
and a5 =
. Find r. (Hint: Use n = 5 in the formula an = a1r n−1 with a1 = 18.)
9
Solution: 32/9 = 18r4 ⇒ (32/9)(1/18) = r4 ⇒ (16/81) = r4 ⇒ r = ± 4
Since the terms are all positive, r =
16
2
=± .
81
3
2
.
3
Suppose we would like to add the terms of a geometric sequence as we did
with the bouncing ball problem at the beginning of this section. Remember that we
added only five terms which was easy, but what if we wanted to add 50 or 100 terms?
If the sequence is geometric, there is a formula that we can use to produce this sum.
SUM OF A GEOMETRIC SEQUENCE
n
Sn = ∑ a1r k−1 =
k=1
a1 (1− r n )
1− r
(r ≠ 1)
EXAMPLE 3
Find the sum of the first 100 terms of the geometric sequence
 1  k−1
given by ak = 6  and show that the answer is very close to 12.
 2

1 
61 − 100 

1 
 2 
Solution: S100 =
= 121 −

1
 1 − 2100 
1−
2
1
1
And since 100 is very close to 0, this means 1 − 100 is very close to 1, so S100 is
2
2
very close to 12.
EXERCISES
1
1. (a) Write the first few terms of the series defined by ∑ 3 
 10 
k=1
values of a1 and r.
k +1
8
1
(b) Use your answer to part (a) to evaluate ∑ 3  .
 10 
k=1
8
k +1
. Note the
2. Suppose that you save $209.92 in January 2005 and that each month thereafter
you only manage to save half of what you saved the previous month. How much
do you save in the tenth month, and what are your total savings after 10 months?
1
of the words are removed
5
each time the paper is rewritten, what is the general term of the sequence for the
number of words remaining in the paper? Use your TI and the general term to
determine the number of words remaining after five rewrites.
3. An initial draft of a term paper contains 9155 words. If
The geometric mean (also known as the mean proportional) between two real
numbers a and b is a number g such that a, g, b is a geometric sequence. In other
words, if r is the common ratio, then ar = g and gr = b. Solving for r yields the
following results:
a g
If g is the geometric mean between a and b, then = , so g 2 = ab , and
g b
g = ± ab provided that ab is a real number. For example, the mean proportionals
between 4 and 9 are ±6, and the mean proportionals between -9 and -4 are also ±6,
whereas -4 and 9 have no mean proportional because −4 ⋅ 9 is not a real number.
HOMEWORK PROBLEMS
1. Find the fourteenth term of the geometric sequence
2. For the geometric sequence with a1 = 100 and r =
to find which term is equal to
1 1 1
, , , . . .
8 4 2
1
, use the formula an = a1r n−1
10
1
.
1010
3. Find r for the geometric sequence having a1 = -25 and a5 = -3.24.
Evaluate the series in question 4 – 6.
10
8
 1  k−1
5. ∑ 3 
4. ∑ 2 k−1
 10 
k=1
k=1
7. Find g > 0 such that
6
6.
∑ (−3)
k−2
k=1
1
25
, g,
forms a geometric sequence.
7
63
8. Insert three geometric means between 6 and 1536.
9. Insert four geometric means between 128 and 4.
10. Suppose someone offered you a job for 30 days. In addition, your boss gave
you the choice of payment as follows: Option 1: $100 per day for 30 days;
Option 2: 1¢ for the first day, 2¢ for day two, 4¢ for day three, and so on so
that each day you got double what you got the day before for 30 days. Which
option would you choose?
11. Suppose the amount of money you save in any given month is twice the
amount you saved in the previous month. How much will you have saved at
the end of one year if you save $1 in January? How much will you have saved
in one year if you saved 25¢ in January?
12. A radioactive substance is decaying so that at the end of each month there is
only one-third as much as there was at the beginning of the month. If there
were 75 grams of the substance at the beginning of the year, how much is left
at midyear?