Ch.10 -11 Review
Precalculus Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Write the first four terms of the sequence whose general term is given.
1) a n = 4(3n - 1)
2) a n = (-1)n (n + 9)
2)
Write the first four terms of the sequence defined by the recursion formula.
3) a 1 = -3 and a n = a n-1 - 3 for n ≥ 2
Evaluate the factorial expression.
9!
4)
7! 2!
5)
1)
3)
4)
n(n + 2 )!
(n + 3 )!
5)
Find the indicated sum.
6
6) ∑ (3i - 3)
i = 3
4
7)
∑
(-1)k(k + 11)
k = 1
7
8)
∑
i = 4
i!
(i - 1 )!
Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index of
summation.
9) 5 + 6 + 7 + 8 + . . . + 31
9)
5
7
19
10) 2 + + 3 + + . . . + 2
2
2
10)
Find the common difference for the arithmetic sequence.
11) 6, 11, 16, 21, . . .
11)
Write the first five terms of the arithmetic sequence.
12) a 1 = -15; d = 3
12)
1
Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence
with the given first term, a 1 , and common difference, d.
13)
13) Find a 13 when a 1 = 29, d = -3.
Write a formula for the general term (the nth term) of the arithmetic sequence. Then use the formula for a n to find a 20,
the 20th term of the sequence.
14) -1 , 1 , 3 , 5 , 7 , . . .
14)
1
3
15) a 1 = - , d = 4
4
15)
Find the indicated sum.
16) Find the sum of the first 20 terms of the arithmetic sequence: -12, -6, 0, 6, . . .
16)
Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
49
17) ∑ (3i + 2)
i = 1
If the given sequence is a geometric sequence, find the common ratio.
18) 4, -12, 36, -108, 324
19)
3 3 3
3
3
, , , , 3 12 48 192 768
18)
19)
Write the first five terms of the geometric sequence.
1
20) a 1 = 6; r = 3
20)
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence
with the given first term, a 1 , and common ratio, r.
21) Find a 11 when a 1 = 4, r = -2.
21)
Write a formula for the general term (the nth term) of the geometric sequence.
22) 6, -12, 24, -48, 96, . . .
22)
The general term of a sequence is given. Determine whether the given sequence is arithmetic, geometric, or neither. If
the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.
23) a n = 4n - 2
23)
24) a n = 3 n
2
24)
Use the formula for the sum of the first n terms of a geometric sequence to solve.
3 3 3
25) Find the sum of the first five terms of the geometric sequence: , , , . . . .
2 8 32
2
25)
Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
5
26) ∑ 4 · 2 i
3
i = 1
Find the sum of the infinite geometric series, if it exists.
1
1
1
27) 1 + + + + . . .
4 16 64
28)
27)
1
- 1 + 3 - . . .
3
28)
A statement Sn about the positive integers is given. Write statements S1 , S2 , and S3 , and show that each of these
statements is true.
n(n + 1)(n + 2)
29)
29) Sn : 1 · 2 + 2 · 3 + 3 · 4 + . . . + n(n + 1) = 3
Use mathematical induction to prove that the statement is true for every positive integer n.
30) 8 + 16 + 24 + . . . + 8n = 4n(n + 1)
31) 1 · 6 + 2 · 6 + 3 · 6 + . . . + 6n = 6n(n + 1)
2
30)
31)
Evaluate the given binomial coefficient.
11
32)
4
32)
Use the Binomial Theorem to expand the binomial and express the result in simplified form.
33) (x + 4y) 3
34) (x2 - 2y) 4
33)
34)
Write the first three terms in the binomial expansion, expressing the result in simplified form.
35) (x + 6 )20
35)
Find the term indicated in the expansion.
36) (2x - 3y) 10; 4th term
36)
37) (2x + 4)5 ; 5th term
37)
Complete the table for the function and find the indicated limit.
x4 - 1
38) lim
x→1 x - 1
x 0.9
f(x)
0.99
0.999 1.001 1.01
1.1
3
38)
The graph of a function is given. Use the graph to find the indicated limit and finction value, or state that the limit or
function value does not exist.
b. f(1)
39)
39) a. lim f(x)
x → 1
2 y
1
-3
-2
-1
1
2
x
-1
-2
-3
-4
Graph the function. Then use your graph to find the indicated limit.
x2 - 16
, lim f(x)
40) f(x) = x + 4
x→-4
y
x
4
40)
The graph of a function is given. Use the graph to find the indicated limit and finction value, or state that the limit or
function value does not exist.
b. f(1)
41)
41) a. lim f(x)
x→1
y
5
4
3
2
1
-2
-1
1
2
3
4
5
x
-1
-2
42) a. lim f(x)
x→1 +
b. f(1)
42)
y
5
4
3
2
1
-2
-1
1
2
3
4
5
x
-1
-2
Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties
can be applied.
x3 - 6x + 8
43)
43) lim
x - 2
x→0
x4 - 1
44) lim
x→1 x - 1
45) lim
x→4
44)
x - 4
x - 2
45)
Determine for what numbers, if any, the given function is discontinuous.
2x + 5
46) f(x) = x2 - 4
5
46)
Find the slope-intercept equation of the tangent line to the graph of f at the given point.
47) f(x) = x2 + 5x at (4, 36)
Find the derivative of f at x. That is, find f ʹ(x).
-4
48) f(x) = ; x = 4
x
47)
48)
Solve the problem.
49) The function f(x) = x2 describes the area of a square, f(x), in square centimeters, whose
sides each measure x centimeters. If x is changing,
(i) Find the average rate of change of the area with respect to x as x changes from 9
centimeters to 9.1 centimeters.
(ii) Find the average rate of change of the area with respect to x as x changes from 9
centimeters to 9.01 centimeters.
(iii) Find the instantaneous rate of change of the area with respect to x at the moment
when x = 9 centimeters.
49)
50) A foul tip of a baseball is hit straight upward from a height of 4 feet with an initial velocity
of 80 feet per second. The function s(t) = -16t2 + 80t describes the ballʹs height above the
50)
ground, s(t), in feet, t seconds after it was hit. The ball reaches its maximum height above
the ground when the instantaneous velocity reaches zero. After how many seconds does
the ball reach its maximum height?
6
Answer Key
Testname: CH.10 REVIEW
1)
2)
3)
4)
8, 20, 32, 44
-10, 11, -12, 13
-3, -6, -9, -12
36
n
5)
n + 3
6) 42
7) 2
8) 22
28
9) ∑ (k + 3)
k = 2
19
k
10) ∑
2
k = 4
11) 5
12) -15, -12, -9, -6, -3
13) -7
14) a n = 2n - 3; a 20 = 37
3
15) a n = n - 1; a 20 = 14
4
16) 900
17) 3773
18) -3
1
19)
4
2 2 2
20) 6, 2, , , 3 9 27
21) 4096
22) a n = 6(-2)n - 1
23) arithmetic, d = 4
3
24) geometric, r = 2
25)
1023
512
26)
248
3
27)
4
3
28) does not exist
7
Answer Key
Testname: CH.10 REVIEW
?
1(1 + 1)(1 + 2)
29) S1 : 1 · 2 =
3
?
1 · 2 · 3
2 =
3
2 = 2 ✓
?
2(2 + 1)(2 + 2)
S2 : 1 · 2 + 2 · 3 =
3
?
2 · 3 · 4
8 =
3
8 = 8 ✓
?
3(3 + 1)(3 + 2)
S3 : 1 · 2 + 2 · 3 + 3 · 4 =
3
?
3 · 4 · 5
20 =
3
20 = 20 ✓
?
30) S1 : 8 = (4 · 1)(1 + 1)
?
8 = 4 · 2
8 = 8 ✓
Sk: 8 + 16 + 24 + . . . + 8k = 4k(k + 1)
Sk+1 : 8 + 16 + 24 + . . . + 8(k + 1) = 4(k + 1)(k + 2)
We work with Sk. Because we assume that Sk is true, we add the next multiple of 8, namely
8(k+1), to both sides.
8 + 16 + 24 + . . . + 8k + 8(k + 1) = 4k(k + 1) + 8(k + 1)
8 + 16 + 24 + . . . + 8(k + 1) = (k + 1)(4k + 8)
8 + 16 + 24 + . . . + 8(k + 1) = 4(k + 1)(k + 2)
We have shown that if we assume that Sk is true, and we add (8(k+1) to both sides of Sk, then Sk+1 is also true. By the
principle of mathematical induction, the statement Sn is true for every positive integer n.
8
Answer Key
Testname: CH.10 REVIEW
?
(6 · 1)(1 + 1)
31) S1 : 1 · 6 =
2
?
6 · 2
6 =
2
6 = 6 ✓
Sk: 1 · 6 + 2 · 6 + 3 · 6 + . . . + 6k = 6k(k + 1)
2
Sk+1 : 1 · 6 + 2 · 6 + 3 · 6 + . . . + 6(k + 1) = 6(k + 1)(k + 2)
2
We work with Sk. Because we assume that Sk is true, we add the next consecutive term, namely
6(k+1), to both sides.ʺ
1 · 6 + 2 · 6 + 3 · 6 + . . . + 6k + 6(k + 1) = 6k(k + 1)
+ 6(k + 1)
2
1 · 6 + 2 · 6 + 3 · 6 + . . . + 6(k + 1) = 6k(k + 1) 12(k + 1)
+ 2
2
1 · 6 + 2 · 6 + 3 · 6 + . . . + 6(k + 1) = (k + 1)(6k + 12)
2
1 · 6 + 2 · 6 + 3 · 6 + . . . + 6(k + 1) = 6(k + 1)(k + 2)
2
We have shown that if we assume that Sk is true, and we add (6(k+1) to both sides of Sk , then Sk+1 is also true. By
the principle of mathematical induction, the statement Sn is true for every positive integer n.
32) 330
33) x3 + 12x2 y + 48xy2 + 64y3
34) x8 - 8x6 y + 24x4 y2 - 32x2 y3 + 16y4
35) x20 + 120 x19 + 6840 x18
36) -414,720 x7 y3
37) 2560x
38) 3.439; 3.940; 3.994; 4.006; 4.060; 4.641
limit = 4.0
39) a. lim f(x) = -3
x → 1
b. f(1) = 1
40) -8
41) a. lim f(x) does not exist
x→1
b. f(1) = 2
42) a. lim f(x) = 2
x→1 +
b. f(1) = 2
43) -4
44) 4
45) 4
46) -2 and 2
47) y = 13x - 16
9
Answer Key
Testname: CH.10 REVIEW
48)
1
4
49) (i) 18.1 square centimeters per centimeter;
(ii) 18.01 square centimeters per centimeter;
(iii) 18 square centimeters per centimeter
5
50) seconds
2
10