Deck 7: Sequences, Series, and Limits

Question 1

Evaluate $\lim _{n \rightarrow \infty} \frac{2 n+4}{3 n-7}$ . Give the exact answer.

Answer 1

$\frac{2}{3}$

Question 2

Evaluate $\lim _{n \rightarrow \infty} \frac{5 n^{2}+6 n}{4 n^{2}-12 n}$ . Give the exact answer.

Answer 2

$\frac{5}{4}$

Question 3

Evaluate $\lim _{n \rightarrow \infty} \frac{11 n-1}{13 n+5}$ . Give the exact answer.

Answer 3

$\frac{11}{13}$

Question 4

Evaluate $\lim _{n \rightarrow \infty} \frac{7 n^{2}+6 n+8}{8 n^{2}+7 n-9}$ . Give the exact answer.

Question 5

Evaluate $\lim _{n \rightarrow \infty} \frac{3 n^{2}-4 n+1}{2 n^{3}+n+4}$ .

A) 2/3
B) 0
C) -4
D) 3/2

Question 6

Evaluate $\lim _{n \rightarrow \infty} \frac{8 n^{2}-2 n+1}{5 n^{2}+3 n-2}$ .

A) $\frac{8}{9}$
B) $\frac{8}{7}$
C) $\frac{8}{5}$
D) $\frac{6}{5}$

Question 7

is $\lim _{n \rightarrow \infty}\left(1+\frac{2}{n}\right)^{n}=e^{2}$ correct?

Question 8

Find the sum of all three-digit positive integers.

Question 9

Find the sum of all four-digit positive integers whose last digit equals 4.

Question 10

Evaluate the geometric series. Leave the answer in terms of unsimplified powers of the common ratio.
$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots+\frac{1}{2^{54}}-\frac{1}{2^{55}}$

Question 11

Evaluate $\lim _{n \rightarrow \infty}\left(1+\frac{8}{n}\right)^{3 n-1}$ .

A) $e^{512}$
B) $e^{24}$
C) $e^{-24}$
D) 1

Question 12

Evaluate $\lim _{n \rightarrow \infty} \frac{n^{2}}{10}\left(e^{\frac{9}{n^{2}}}-1\right)$ .

Question 13

Evaluate $\lim _{n \rightarrow \infty} n \ln \left(1+\frac{1}{10 n}\right)$ .

Question 14

Evaluate $\lim _{n \rightarrow \infty} n\left[\ln \left(7+\frac{1}{n}\right)-\ln 7\right]$ .

Question 15

is $\lim _{n \rightarrow \infty} 3 n \ln \left(1+\frac{1}{4 n}\right)=1-\frac{1}{4}$ correct?

Question 16

Evaluate $\lim _{n \rightarrow \infty} 6 n\left[\ln \left(5+\frac{1}{6 n}\right)-\ln 5\right]$ .

A) $\frac{1}{5}$
B) $\frac{1}{6}$
C) $\frac{5}{6}$
D) $\frac{1}{30}$

Question 17

Evaluate $\sum_{m=1}^{\infty} \frac{4}{6^{m}}$ .

Question 18

Evaluate $\sum_{m-2}^{\infty} \frac{1}{2^{m}}$ .

Question 19

is $\sum_{m=7}^{\infty} \frac{1}{8^{m}}=\frac{1}{14680064}$ correct?

Question 20

Evaluate $\sum_{m=4}^{\infty} \frac{7}{3^{m}}$ .

A) $\frac{7}{81}$
B) $\frac{7}{54}$
C) $\frac{7}{27}$
D) $\frac{7}{162}$

Question 21

Express 31.14 65 65 65 as a fraction; here the digits 65 keep repeating forever.

Question 22

Express $0 . \overline{94}$ as a fraction.

Question 23

Express $0.1 \overline{7}$ as a fraction.

Question 24

Express $0 . \overline{406}$ as a fraction.

Question 25

Express $0.39 \overline{4}$ as a fraction.

Question 26

Evaluate the arithmetic series.
93 + 94 + 95 + ... + 306 + 307

A) 42,785
B) 43,000
C) 43,215
D) 42,570

Question 27

Evaluate the arithmetic series.
44 + 51 + 58 + ... + 926 + 933

Question 28

Evaluate the arithmetic series.
100 + 115 + 130 + ... + 3280 + 3295

Question 29

Evaluate the arithmetic series.
$200+190+180+\cdots-140-150$

Question 30

Evaluate the arithmetic series.
$\sum_{k=1}^{\infty}(3+7 k)$

Question 31

Evaluate the arithmetic series. $\sum_{k=8}^{500}(4 k-8)$

A) 497,056
B) 497,064
C) 496,944
D) 496,952

Question 32

Find the sum of all three-digit positive integers whose last digit equals 3.

Question 33

Evaluate the geometric series.
1 + 5 + 25 + ... + $5^{170}$

A) $\frac{5^{172}-1}{4}$
B) $\frac{5^{169}-1}{4}$
C) $\frac{5^{170}-1}{4}$
D) $\frac{5^{171}-1}{4}$

Question 34

Evaluate the geometric series. Leave the answer in terms of unsimplified powers of the common ratio.
5 + 10 + 20 + ... + $5 \cdot 2^{200}$

Question 35

Evaluate the geometric series. Leave the answer in terms of unsimplified powers of the common ratio.
$1+\frac{1}{6}+\frac{1}{36}+\cdots+\frac{1}{6^{52}}$

Question 36

Evaluate the geometric series. $1-\frac{1}{4}+\frac{1}{16}-\frac{1}{64}+\cdots+\frac{1}{4^{66}}-\frac{1}{4^{67}}+\frac{1}{4^{68}}$

A) $\frac{4^{69}+1}{5 \cdot 4^{68}}$
B) $\frac{4^{69}+1}{4^{68}}$
C) $\frac{4^{68}+1}{5 \cdot 4^{67}}$
D) $\frac{4^{68}+1}{4^{67}}$

Question 37

Evaluate the geometric series. Leave the answer in terms of unsimplified powers of the common ratio.
$\sum_{k=1}^{59} \frac{8}{3^{k}}$

Question 38

Evaluate the geometric series. Leave the answer in terms of unsimplified powers of the common ratio.
$\sum_{k=1}^{83}(-3)^{k}$

Question 39

A) Write the following series explicitly.
B) Evaluate the sum.
$\sum_{k=1}^{4}\left(k^{2}+2\right)$

Question 40

Write the series explicitly and evaluate the sum. Use log rules to simply your answer.
$\sum_{k=0}^{5} \ln \left(4+3^{k}\right)$

Question 41

Write the series explicitly and evaluate the sum. Give the exact answer.

$\sum_{k=2}^{4}\left(\sin \frac{\pi}{k}+\cos \frac{\pi}{k}\right)$

Question 42

Find the sum of all three digit even positive integers.

Question 43

Write the series using summation notation (starting with m = 1).
8 + 11 + 14 + ... + 167

Question 44

Write the series using summation notation (starting with m = 1).
$\frac{3}{4}+\frac{3}{16}+\frac{3}{64}+\cdots+\frac{3}{2^{30}}$

Question 45

Write the series using summation notation (starting with m = 1).
$-\frac{8}{3}+\frac{8}{9}-\frac{8}{27}+\cdots-\frac{8}{3^{65}}+\frac{8}{3^{66}}$

A) $\sum_{m=1}^{66} \frac{8}{(-3)^{m}}$
B) $528 \cdot \sum_{m=1}^{66}\left(-\frac{1}{3}\right)^{m}$
C) $\sum_{m=1}^{65} \frac{8}{(-3)^{m+1}}$
D) $\sum_{m=1}^{66}(-1)^{m} \frac{8 m}{3^{m}}$

Question 46

Suppose you start an exercise program by riding your bicycle 15 miles on the first day and then you increase the distance you rode by 0.25 miles each day. How many total miles did you ride after 45 days?

Question 47

Consider the fable where one grain of rice is placed on the first square of a chessboard, then two grains on the square, then four grains on the third square, and so on, doubling the number of grains placed on each square. Find the total number of grains of rice on the first 15 squares of the chessboard.

Question 48

Find the 5th row of the Pascal's Triangle.

Question 49

Evaluate
$\binom{8}{3}$

Question 50

Find the coefficient of x²⁷ in the expansion of (x + 5)³⁰.

Question 51

Give the first four terms of the sequence $a_{n}=\sqrt{\frac{n}{n+11}}$ . Give the exact answers.

Question 52

Give the first four terms of the sequence a_n = 2 + 2ⁿ.

Question 53

Find the 101th term of the sequence $a_{n}=\sqrt{\frac{2 n-1}{3 n-2}}$ . Give the exact answer.

Question 54

Give the first four terms of the arithmetic sequence whose first term is 7 and whose difference between consecutive terms is 7.

Question 55

Which of the following arithmetic sequences has third term 7?

A) First term 4; difference 3
B) First term 1; difference 3
C) First term -2; difference 6
D) First term -5; difference 7

Question 56

Find two numbers between 6 and 18 such that they form an arithmetic sequence with 6 and 18.

Question 57

Give the first four terms of a geometric sequence whose first term is 9 and ratio 2 of consecutive terms.

Question 58

Find the eighth term of an arithmetic sequence whose third term is 7 and whose fifth term is 13.

Question 59

Which of the following geometric sequences has fourth term a⁵?

A) First term $\frac{1}{a}$ ; ratio a²
B) First term a; ratio a
C) First term $\frac{1}{a^{2}}$ ; ratio a²
D) First term 1; ratio a²

Question 60

Find the nth term of an arithmetic sequence whose difference is 3 and whose (n - 3)th term is 2.

Question 61

Find the first four terms of the recursive sequence a₁ = 3, a_{n + 1} = 3a_n - 2 for n ≥ 1.

Question 62

Find the fifth term of the recursive sequence a₁ = a, a₂ = b, a_{n + 2} = a_n*a_{n + 1} for n $\ge$ 1.

A) ab²
B) ab³
C) a²b²
D) a²b³

Question 63

Define a recursive sequence using the equations a₁ = 7,
$a_{n+1}=\left\{\begin{array}{ll}\frac{a_{n}}{2}, & \text { if } n \text { is even } \\ 2 a_{n}+3, & \text { if } n \text { is odd }\end{array}\right.$
.
Find the smallest value of n such that a_n = 10.

Question 64

Find the nth term of a recursive sequence given by a₁ = 5, a_{n + 1} = a_n + 4 for n $\ge$ 1.

A) 5 + 4n
B) 5 + 4(n + 1)
C) 5 + 4(n - 1)
D) 9n

Question 65

Consider the sequence whose n^th term is given by $a_{n}=4 n-2$ . Write the sequence as a recursive sequence.

A) $a_{1}=2, a_{n+1}=a_{n}+2$
B) $a_{1}=2, a_{n+1}=a_{n}+4$
C) $a_{1}=2, a_{n+1}=a_{n}-2$
D) $a_{1}=2, a_{n+1}=a_{n}-4$

Question 66

Find two numbers between 5 and 40 such that they form a geometric sequence with 5 and 40.

Question 67

If $a_{7}=3$ and $a_{11}=19$ are terms of an arithmetic sequence, find the difference between consecutive terms of this sequence.

Question 68

Find the first term of an arithmetic sequence such that $a_{3}=8$ and $a_{4}=13$ .

Question 69

Find the 5th term of a sequence defined by $a_{n}=\frac{(-1)^{n}}{n}$ .

Question 70

Find the first term of an arithmetic sequence whose second term is 23 and whose fourth term is 28.

A) 18.5
B) 19.5
C) 20.5
D) 21.5

Question 71

At the first day of a new year you have 1,017 e-mail messages saved on your computer. At the end of each day you save only your 14 most important new e-mail messages. How many messages will you have saved on the 97th day of the year?

Question 72

Suppose your annual salary at the beginning of your first year at a new company is $40,000. Assume your salary increases by 9% per year at the end of each year of employment. What is your salary at the end of the 6th year? Round the answer to the nearest dollar.