Chapter 12 Sequences; Induction; the Binomial Theorem

Chapter 12: Sequences; Induction; the Binomial Theorem
57. The total number of seats is:
S = 25 + 26 + 27 + + ( 25 + 29()
1 )
This is the sum of an arithmetic sequence with
d = 1, a1
= 25, and n= 30 .
Find the sum of the sequence:
30
S 30 = [ 2(25) + (30 − 1)(1) ]
2
= 15(50 + 29) = 15(79)
= 1185
There are 1185 seats in the theater.
58. The total number of seats is:
( ( ( )))
S = 15 + 17 + 19 + + 15 + 39 2
This is the sum of an arithmetic sequence with
d = 2, a1
= 15, and n = 40 .
Find the sum of the sequence:
40
S 40 = [ 2(15) + (40 − 1)(2) ]
2
= 20(30 + 78) = 20(108)
= 2160
The corner section has 2160 seats.
59. The lighter colored tiles have 20 tiles in the
bottom row and 1 tile in the top row. The
number decreases by 1 as we move up the
triangle. This is an arithmetic sequence with
a1 = 20, d =− 1, and n = 20 . Find the sum:
20
S = [ 2(20) + (20 − 1)( − 1) ]
2
= 10(40 − 19) = 10(21)
= 210
There are 210 lighter tiles.
The darker colored tiles have 19 tiles in the
bottom row and 1 tile in the top row. The
number decreases by 1 as we move up the
triangle. This is an arithmetic sequence with
a1 = 19, d =− 1, and n = 19 . Find the sum:
19
S = [ 2(19) + (19 − 1)( − 1) ]
2
19 19
= (38 − 18) = (20) = 190
2 2
There are 190 darker tiles.
60. The number of bricks required decreases by 2 on
each successive step. This is an arithmetic
sequence with a1 = 100, d =− 2, and n= 30 .
a. The number of bricks for the top step is:
a30 = a1 + ( n− 1) d = 100 + (30 −1)( −2)
= 100 + 29( − 2) = 100 −58
= 42
42 bricks are required for the top step.
b. The total number of bricks required is the
sum of the sequence:
30
S = [ 100 + 42 ] = 15(142) = 2130
2
2130 bricks are required to build the
staircase.
61. The air cools at the rate of 5.5° F per 1000 feet.
Since n represents thousands of feet, we have
d = − 5.5 . The ground temperature is 67° F so
we have T 1 = 67 − 5.5 = 61.5 . Therefore,
T = 61.5 + n−1 −5.5
{ }
{ n} ( )( )
= { − 5.5n+ 67 } or { 67 −5.5n}
After the parcel of air has risen 5000 feet, we
have T 5 = 61.5 + ( 5 −1)( − 5.5)
= 39.5 .
The parcel of air will be 39.5° F after it has risen
5000 feet.
62. If we treat the length of each rung as the term of
an arithmetic sequence, we have 1 49 a = ,
d = − 2.5 , and a = 24 .
n
n
( )
( n )( )
( n )
a = a + n−
d
1
1
24 = 49 + −1 −2.5
− 25 =−2.5 −1
10 = n −1
11 = n
Therefore, the ladder contains 11 rungs.
To find the total material required for the rungs,
we need the sum of their lengths. Since there are
11 rungs, we have
11 11
S
11
= ( 49 + 24) = ( 73)
= 401.5
2 2
It would require 401.5 feet of material to
construct the rungs for the ladder.
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