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Alg2 - CH12 Practice Test
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Find the first 5 terms of the sequence with a1 = 6 and an = 2an − 1 − 1 for n ≥ 2.
a. 1, 2, 3, 4, 5
c. 6, 12, 24, 48, 96
b. 6, 11, 21, 41, 81
d. 6, 7, 8, 9, 10
____
2. Write the series − 12 +
a. 6
∑ ( −1 )kÊÁÁÁ 2k1 ˆ˜˜˜
Ë
¯
k=1
6
b.
∑ ( −1 )k + 1ÊÁÁÁ 2k1 ˆ˜˜˜
Ë ¯
k=1
1
4
− 16 + 18 −
1
10
+
1
12
in summation notation.
c. 6
∑ ( −1 )kÊÁÁÁ 12 ˆ˜˜˜ k
Ë ¯
k=1
6
d.
∑ ( −1 )k + 1ÊÁÁÁ 12 ˆ˜˜˜ k
Ë ¯
k=1
22
____
3. Evaluate the series ∑ k .
k=1
____
____
____
____
____
____
a. 506
c. 22
b. 253
d. 23
4. Find the 22nd term in the arithmetic sequence –5, –9, –13, –17, –21,...
a. –93
c. –110
b. –84
d. –89
5. Determine whether the sequence 12, 40, 68, 96 could be geometric or arithmetic. If possible, find
the common ratio or difference.
a. It could be geometric with r = 28.
c. It is neither.
b. It could be arithmetic with d = –28.
d. It could be arithmetic with d = 28.
6. Find the geometric mean of − 12 and − 181 .
a. ± 1
b. ± 91
c. ± 1
d. ± 6
36
9
7. There are 256 players competing in a national chess championship tournament. The players
compete until there is 1 winner. How many matches must be scheduled in order to complete the
tournament?
a. 255 matches
c. 128 matches
b. 256 matches
d. 511 matches
8. Find the sum of the infinite geometric series 5 + 5 + 5 + 5 + ..., if it exists.
a. 7.5
c. The
3 ratio
9 diverges.
27
b. 7.41
d. 1
9. Write 0.959595... as a fraction in simplest form.3
a. 95
c. 19
b. 999
d. 200
19
95
20
99
____
10. Use mathematical induction to prove that the sum of the squares of the first n natural numbers is
n
∑k
2
= 12 + 22 + 32 +. . .+ n2 =
k=1
n(n + 1)(2n + 1)
.
6
Complete the proof.
Proof: Base case: Show that the statement is true for n = 1.
[1.]
The base case is true.
Assume that the statement is true for a natural number, k.
[2.]
Replace n with k.
Prove that the statement is true for the natural number k + 1.
12 + 22 + 32 +. . .+ k2 + (k + 1)2 = k(k + 1)(2k + 1) + (k + 1)2
6
k(k + 1)(2k + 1) + 6(k + 1) (k + 1)[k(2k + 1) + 6(k + 1)]
=
=
6
6
2
(k + 1)[2k + k + 6k + 1] (k + 1)(k + 2)(2k + 3) (k + 1)((k + 1) + 1)(2(k + 1) + 1)
=
=
=
6
6
6
2
n
Therefore,
∑k
2
= 12 + 22 + 32 +. . .+ n2 =
k=1
n(n + 1)(2n + 1)
.
6
n(n + 1)(2n + 1) 1(1 + 1)(2(1) + 1) 6
=
= =1
6
6
6
k(k + 1)(2k + 1)
2
2
2
2
2
+ (k + 1)2
[2.] 1 + 2 + 3 +. . .+ k + (k + 1) =
6
b.
n(n
+
1)(2n
+
1)
1(1
+
1)(2(1)
+
1)
=
= 6 =1
[1.] 12 =
6
6
6
k(k + 1)(2k + 1)
[2.] 12 + 22 + 32 +. . .+ k2 =
6
c.
n(n + 1)(2n + 1) 2(2 + 2)(2(2) + 2) 48
=
=
=8
[1.] 12 =
6
6
6
k(k + 1)(2k + 1)
[2.] 12 + 22 + 32 +. . .+ k2 =
6
d.
n(n + 1)(2n + 1) 2(2 + 2)(2(2) + 2) 48
=
=
=8
[1.] 12 =
6
6
6
k(k + 1)(2k + 1)
2
2
2
2
2
+ (k + 1)2
[2.] 1 + 2 + 3 +. . .+ k + (k + 1) =
6
a.
____
[1.] 12 =
11. Identify a counterexample to disprove 6n ≥
a. Counterexample: n = 0
n2 , where n is a real number.
6
c. Counterexample: n = 36
b. Counterexample: n = 6
____
____
12. Estimate the area under the curve f( x )
a. 30 square units
b. 112.5 square units
d. Counterexample: n = 37
= −x2 + 4x + 4.25 over 0 ≤ x ≤ 5. Use 5 intervals.
c. 40.5 square units
d. 80 square units
13. Use the sum of a series to estimate the area under the curve f( x ) = −x2 + 12x + 13 over 0 ≤ x ≤ 12.
Use 4 intervals.
a. 318 square units
c. 534 square units
b. 460 square units
d. 453 square units
Numeric Response
2
2
2
2
14. Find the sum of the series 3 + 6 + 9 + … + 36 .
15. Write the repeating decimal 0.26 as a fraction in simplest form.
Matching
Match each vocabulary term with its definition.
a. explicit formula
b. sequence
c. partial sum
d. series
e. implicit formula
f. summation notation
g. iteration
n
____
16. indicated by Sn =
∑ a , the sum of a specified number of terms n of a sequence whose total
i
i=1
____
____
number of terms is greater than n
17. the indicated sum of the terms of a sequence
18. a formula that defines the nth term an, or general term, of a sequence as a function of n
____
19. the repetitive application of the same rule
Match each vocabulary term with its definition.
a. arithmetic sequence
b. arithmetic series
c. geometric sequence
d. geometric series
e. geometric mean
f. infinite sequence
g. finite sequence
____
____
____
____
20. a sequence whose successive terms differ by the same nonzero number d, called the common
difference
21. the indicated sum of the terms of an arithmetic sequence
22. a sequence in which the ratio of successive terms is a constant r, called the common ratio, where
r ≠ 0 and r ≠ 1
23. the indicated sum of the terms of a geometric sequence
Match each vocabulary term with its definition.
a. geometric mean
b. diverge
c. iteration
d. mathematical induction
e. limit
f. converge
g. infinite geometric series
____
____
____
____
24.
25.
26.
27.
a type of mathematical proof
when the common ratio | r | ≥ 1 and the partial sums do not approach a fixed number
a geometric series with infinitely many terms
when the common ratio | r | < 1 and the partial sums approach a fixed number
Alg2 - CH12 Practice Test
Answer Section
MULTIPLE CHOICE
1. ANS: B
The first term is given a1 = 6.
Substitute this value into the rule to find the next term.
Continue using each term to find the next term.
n
1
2
3
4
5
2an − 1 − 1
an
2(6) − 1
2(11) − 1
2(21) − 1
2(41) − 1
6
11
21
41
81
Feedback
A
B
C
D
The first term is given. Use the rule to find each following term.
Correct!
Remember to use both parts of the rule when finding each of the terms.
You have the correct first term. Use the rule to find the following terms.
PTS: 1
DIF: Average
REF: Page 862
OBJ: 12-1.1 Finding Terms of a Sequence by Using a Recursive Formula
NAT: 12.5.1.a
TOP: 12-1 Introduction to Sequences
2. ANS: A
Find a rule for the kth term.
ˆ
kÊ
ak = ( −1 ) ÁÁÁ 1 ˜˜˜
Ë 2k ¯
Explicit formula
Write the notation for the first 6 items.
6
∑ ( −1 ) ÊÁÁÁ 2k1 ˆ˜˜˜
k
k=1
Ë
¯
Summation notation
Feedback
A
B
C
D
Correct!
Does the sequence start with a minus sign or a plus sign?
Is the whole fraction multiplied by k or just the denominator?
Does the sequence start with a minus sign or a plus sign? Is the whole fraction
multiplied by k or just the denominator?
PTS: 1
DIF: Average
REF: Page 870
OBJ: 12-2.1 Using Summation Notation TOP: 12-2 Series and Summation Notation
3. ANS: B
22
∑ k = n(n2+ 1) = 22(23)
= 253
2
Use the summation formula for a linear
series.
k=1
Feedback
A
B
C
D
Use the correct summation formula.
Correct!
Find the sum of the first n natural numbers, where n is the number on top of the
summation notation.
Use the summation formula for a linear series.
PTS: 1
DIF: Average
REF: Page 872
OBJ: 12-2.3 Using Summation Formulas TOP: 12-2 Series and Summation Notation
4. ANS: D
Find a specific term from a given sequence by using the equation an = a1 + (n − 1)d, where:
an = your result
a1 = the initial term of the sequence
n = the number in the sequence you want to calculate
d = the common difference between the terms
n is given in the problem, a1 is the first term in the sequence, and d is the difference between
adjacent terms.
Feedback
A
B
C
D
Find the correct term in the sequence.
The equation for the nth term contains (n – 1).
Check that the common difference is correct.
Correct!
PTS:
OBJ:
NAT:
KEY:
5. ANS:
1
DIF: Average
REF: Page 880
12-3.2 Finding the nth Term Given an Arithmetic Sequence
12.5.1.a
TOP: 12-3 Arithmetic Sequences and Series
arithmetic sequence | finding given term | nth term
D
12
40
68
96
Difference
28
28
28
Ratio
10
17
24
3
10
It could be arithmetic with d = 28.
17
Feedback
A
B
C
D
Is the ratio or difference of successive terms a constant?
To find the common ratio divide each term by the previous term. To find the
common difference subtract each term from the next term.
Find out if the ratio or difference of successive terms is a constant.
Correct!
PTS:
OBJ:
TOP:
6. ANS:
1
DIF: Basic
REF: Page 890
12-4.1 Identifying Geometric Sequences
12-4 Geometric Sequences and Series
C
Geometric mean formula
± ab
=±
=±
ÊÁ − 1 ˆ˜ÊÁ − 1 ˆ˜
Ë 2 ¯Ë 18 ¯
1
36
NAT: 12.5.1.a
Substitute into the formula.
Multiply.
= ± 16
Simplify.
Feedback
A
B
C
D
Find the square root of the product of the two numbers.
The geometric mean is the term between any two nonconsecutive terms of a
geometric sequence.
Correct!
Find the square root of the product of the two numbers.
PTS:
OBJ:
TOP:
7. ANS:
Step 1
Let
1
DIF: Average
REF: Page 892
12-4.4 Finding Geometric Means NAT: 12.5.1.a
12-4 Geometric Sequences and Series
A
Write a sequence.
n = number of matches played in the nth round.
Sn = total number of matches
n−1
The first round requires 128 matches. Each
an = 128ÊÁË 12 ˆ˜¯
successive match requires 12 as many matches.
Step 2 Find the number of matches required.
n−1
The final round will have 1 match, so substitute
1 = 128ÊÁË 12 ˆ˜¯
1 for an.
n−1
Isolate the exponential expression by dividing
1
ÊÁ 1 ˆ˜
128 = Ë 2 ¯
by 128.
ÊÁ 1 ˆ˜7 = ÊÁ 1 ˆ˜n − 1
Ë2¯ Ë2¯
Express the left side as a power of 12 .
7 = n−1
Equate the exponents.
Step 3
Solve for n.
Find the total number of matches after 8 rounds.
8=n
ÊÁ Ê 1 ˆ8 ˆ˜
ÁÁ 1 − Á ˜ ˜˜
Á
2
˜
Sn = 128ÁÁÁÁ Ë ¯ ˜˜˜˜ = 255 Sum function for geometric series
ÁÁ 1 − ÊÁ 12 ˆ˜ ˜˜
Á
Ë ¯ ˜¯
Ë
255 matches must be scheduled to complete the tournament.
Feedback
A
B
C
D
Correct!
Determine how many rounds will be needed, then use the sum function for a
geometric series to find the total number of matches.
Determine how many rounds will be needed, then use the sum function for a
geometric series to find the total number of matches.
Determine how many rounds will be needed, then use the sum function for a
geometric series to find the total number of matches.
PTS: 1
DIF: Average
REF: Page 894
OBJ: 12-4.6 Application
NAT: 12.5.1.a
TOP: 12-4 Geometric Sequences and Series
8. ANS: A
Find the constant ratio: r = 1 .
| r | < 1 so the series does converge.
3
Use the sum formula: S =
a1
= 7.5.
1−r
Feedback
A
B
C
D
Correct!
This is the sum of the first 4 terms. Find the sum of all the terms.
The ratio is between –1 and 1 so the ratio sum converges.
This is the ratio; use it in the sum formula to find the sum.
PTS:
OBJ:
TOP:
9. ANS:
1
DIF: Average
REF: Page 901
12-5.2 Finding the Sums of Infinite Geometric Series
12-5 Mathematical Induction and Infinite Geometric Series
D
0.959595...
Use the pattern for the series.
= 0.95 + 0.0095 + 0.000095 +. . .
r = 0.0095 = 1 , or 0.01
0.95
100
S=
a1
= 0.95 = 0.95 = 95
1 − r 1 − 0.01 0.99 99
S = 95
99
| r | < 1; the series converges to a sum.
Apply the sum formula.
Simplify if necessary.
Feedback
A
B
C
D
Check by dividing the fraction using a calculator. Do you get the repeating
decimal?
Write the repeating decimal as an infinite geometric series, and apply the sum
formula.
Write the repeating decimal as an infinite geometric series, and apply the sum
formula.
Correct!
PTS: 1
DIF: Average
REF: Page 902
OBJ: 12-5.3 Writing Repeating Decimals as Fractions
TOP: 12-5 Mathematical Induction and Infinite Geometric Series
10. ANS: B
Step 1 Base case: Show that the statement is true for n = 1.
12 =
n(n + 1)(2n + 1) 1(1 + 1)(2(1) + 1) 6
=
= =1
6
6
6
The base case is true.
Step 2 Assume that the statement is true for a natural number, k.
12 + 22 + 32 +. . .+ k2 =
k(k + 1)(2k + 1)
6
Replace n with k.
Step 3 Prove that the statement is true for the natural number k + 1.
k(k + 1)(2k + 1)
+ (k + 1)2
6
k(k + 1)(2k + 1) + 6(k + 1)2 (k + 1)[k(2k + 1) + 6(k + 1)]
=
=
6
6
2
(k + 1)[2k + k + 6k + 1] (k + 1)(k + 2)(2k + 3)
=
=
6
6
(k + 1)((k + 1) + 1)(2(k + 1) + 1)
=
6
12 + 22 + 32 +. . .+ k2 + (k + 1)2
=
n
Therefore,
∑k
k=1
2
= 12 + 22 + 32 +. . .+ n2 =
n(n + 1)(2n + 1)
.
6
Feedback
A
B
C
D
Replace n with k to show that the statement is true for a natural number k.
Correct!
Show that the statement is true for n = 1.
Show that the statement is true for n = 1.
PTS: 1
DIF: Average
REF: Page 902
OBJ: 12-5.4 Proving with Mathematical Induction
TOP: 12-5 Mathematical Induction and Infinite Geometric Series
11. ANS: D
If n = 37 then the statement is false.
2
6(37) ≥ 37
6
222 ≥ 228.17 false.
Feedback
A
B
C
D
The statement is true for this value.
The statement is true for this value.
The statement is true for this value.
Correct!
PTS: 1
DIF: Average
REF: Page 903
OBJ: 12-5.5 Using Counterexamples
TOP: 12-5 Mathematical Induction and Infinite Geometric Series
12. ANS: A
Graph the function. Divide the area into 5 rectangles, each with a width of 1 unit.
x
f(x)
0.5
6
1.5
8
2.5
8
3.5
6
4.5
2
Find the height of each rectangle by evaluating the function at the center of each rectangle, as
shown in the table.
Approximate the area by finding the sum of the areas of the rectangles.
A ≈ 1( 6 ) + 1( 8 ) + 1( 8 ) + 1( 6 ) + 1( 2 ) = 30
The estimate of 30 square units is very close to the actual area of 29 127 square units.
Feedback
A
Correct!
B
C
D
Square x before you multiply by (–1).
Graph the function and divide the area into 5 rectangles, each with a width of 1
unit. Then, approximate the area by finding the sum of the areas of the
rectangles.
Graph the function and divide the area into 5 rectangles, each with a width of 1
unit. Then, approximate the area by finding the sum of the areas of the
rectangles.
PTS: 1
DIF: Average
REF: Page 910
OBJ: 12-Ext.1 Finding Area Under a Curve
TOP: 12-Ext Area Under a Curve
13. ANS: D
Step 1 Graph the function.
Step 2 Divide the area into 4 rectangles, each with a width of 3 units.
Step 3 Find the value of the function at the center of each rectangle, as shown in the table.
ak
f ÊÁË ak ˆ˜¯
a1 = 1.5
a2 = 4.5
a3 = 7.5
a4 = 10.5
28.75
46.75
46.75
28.75
Step 4 Write the sum that approximates the area.
4
ÈÍ
˘˙
A ≈ 3 ∑ f ÁÊË ak ˜ˆ¯ = 3ÍÍÍ f ÁÊË a1 ˜ˆ¯ + f ÁÊË a2 ˜ˆ¯ + f ÁÊË a3 ˜ˆ¯ + f ÁÊË a4 ˜ˆ¯ ˙˙˙
Î
˚
k=1
= 3[ 28.75 + 46.75 + 46.75 ] + 28.75 = 453
The estimated area is 453 square units.
Feedback
A
B
C
Multiply the value of the function at the center of each rectangle by the width of
the rectangle.
Divide the area into 4, not 3, rectangles, each with a width of 3 units.
Multiply the value of the function at the center of each rectangle by the width of
the rectangle.
D
Correct!
PTS: 1
DIF: Average
REF: Page 910
OBJ: 12-Ext.2 Finding Area Under a Curve by Using a Series
TOP: 12-Ext Area Under a Curve
NUMERIC RESPONSE
14. ANS: 5850
PTS:
15. ANS:
1
DIF:
Advanced
TOP: 12-2 Series and Summation Notation
4
15
PTS: 1
DIF: Average
TOP: 12-5 Mathematical Induction and Infinite Geometric Series
MATCHING
16. ANS:
TOP:
17. ANS:
TOP:
18. ANS:
TOP:
19. ANS:
TOP:
C
PTS: 1
DIF:
12-2 Series and Summation Notation
D
PTS: 1
DIF:
12-2 Series and Summation Notation
A
PTS: 1
DIF:
12-1 Introduction to Sequences
G
PTS: 1
DIF:
12-1 Introduction to Sequences
Basic
REF: Page 870
Basic
REF: Page 870
Basic
REF: Page 863
Basic
REF: Page 864
20. ANS:
TOP:
21. ANS:
TOP:
22. ANS:
TOP:
23. ANS:
TOP:
A
PTS: 1
DIF:
12-3 Arithmetic Sequences and Series
B
PTS: 1
DIF:
12-3 Arithmetic Sequences and Series
C
PTS: 1
DIF:
12-4 Geometric Sequences and Series
D
PTS: 1
DIF:
12-4 Geometric Sequences and Series
Basic
REF: Page 879
Basic
REF: Page 882
Basic
REF: Page 890
Basic
REF: Page 893
24. ANS:
TOP:
25. ANS:
TOP:
26. ANS:
TOP:
27. ANS:
TOP:
D
PTS: 1
DIF: Basic
REF:
12-5 Mathematical Induction and Infinite Geometric Series
B
PTS: 1
DIF: Basic
REF:
12-5 Mathematical Induction and Infinite Geometric Series
G
PTS: 1
DIF: Basic
REF:
12-5 Mathematical Induction and Infinite Geometric Series
F
PTS: 1
DIF: Basic
REF:
12-5 Mathematical Induction and Infinite Geometric Series
Page 902
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