䊳
WORKING WITH FORMULAS
73. Sum of the first n natural numbers:
The sum of the first n natural numbers can be
found using the formula shown, where n represents
the number of terms in the sum. Verify the formula
by adding the first six natural numbers by hand,
and then evaluating S
6
. Then find the sum of the
first 75 natural numbers.
S
n
ⴝ
n1n ⴙ 12
2
74. Sum of the squares of the first n natural
numbers:
If the first n natural numbers are squared, the sum
of these squares can be found using the formula
shown, where n represents the number of terms in
the sum. Verify the formula by computing the sum
of the squares of the first six natural numbers by
hand, and then evaluating S
6
. Then find the sum of
the squares of the first 20 natural numbers:
11
2
⫹ 2
2
⫹ 3
2
⫹
p
⫹ 20
2
2.
S
n
ⴝ
n1n ⴙ 1212n ⴙ 12
6
䊳
APPLICATIONS
75. Temperature fluctuation:At 5
P
.
M
. in Coldwater,
the temperature was a chilly If the temperature
decreased by every half-hour for the next 7 hr, at
what time did the temperature hit
76. Arc of a baby swing: When Mackenzie’s baby
swing is started, the first swing (one way) is a 30-in.
arc. As the swing slows down, each successive arc
is in. less than the previous one. Find (a) the length
of the tenth swing and (b) how far Mackenzie has
traveled during the 10 swings.
77. Computer animations: The animation on a new
computer game initially allows the hero of the
game to jump a (screen) distance of 10 in. over
booby traps and obstacles. Each successive jump
is limited to in. less than the previous one. Find
(a) the length of the seventh jump and (b) the total
distance covered after seven jumps.
78. Seating capacity:
The Fox Theater
creates a “theater in
the round” when it
shows any of
Shakespeare’s
plays. The first row
has 80 seats, the
second row has 88,
3
4
3
2
0°F?
3°F
36°F.
the third row has 96, and so on. How many seats
are in the 10th row? If there is room for 25 rows,
how many chairs will be needed to set up the
theater?
79. Sales goals: At the time that I was newly hired,
100 sales per month was what I required. Each
following month—the last plus 20 more, as I work
for the goal of top sales award. When 2500 sales
are thusly made, it’s Tahiti, Hawaii, and piña
coladas in the shade. How many sales were made
by this person in the seventh month? What were
the total sales after the 12th month? Was the goal
of 2500 total sales met?
80. Bequests to charity: At the time our mother left this
Earth, she gave $9000 to her children of birth. This
we kept and each year added $3000 more, as a
lasting memorial from the children she bore. When
$42,000 is thusly attained, all goes to charity that
her memory be maintained. What was the balance in
the sixth year? In what year was the goal of $42,000
met?
Use the alternative formula for the nth partial sum to compute the sums indicated.
67. The sum S
15
for the sequence
68. The sum S
20
for the sequence
69. The sum S
30
for the sequence
0.003 ⫹ 0.173 ⫹ 0.343 ⫹ 0.513 ⫹
p
9
2
⫹
7
2
⫹
5
2
⫹
3
2
⫹
p
⫺12 ⫹ 1⫺9.52⫹ 1⫺72⫹ 1⫺4.52⫹
p
70. The sum S
50
for the sequence
71. The sum S
20
for the sequence
72. The sum S
10
for the sequence
1213
⫹ 1013 ⫹ 813 ⫹ 613 ⫹
p
12
⫹ 212 ⫹ 312 ⫹ 412 ⫹
p
1⫺22⫹ 1⫺72⫹ 1⫺122⫹ 1⫺172⫹
p
9–21 Section 9.2 Arithmetic Sequences 781
College Algebra Graphs & Models—
cob19545_ch09_774-782.qxd 12/17/10 12:48 AM Page 781