Question 1

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10; &#10;A) 5, 0, -5, -10, -15&#10;B) 5, -25, 125, -625, 3125&#10;C) -5, 5, -5, 5, -5&#10;D) 5, -5, 5, -5, 5

Accepted Answer

We can use the recursion formula to find the first few terms of the sequence:
$a_1 = 5$
$a_2 = -a_1 = -5$
$a_3 = -a_2 = 5$
$a_4 = -a_3 = -5$
$a_5 = -a_4 = 5$
Thus, the sequence alternates between 5 and -5, so the answer is choice D.

Question 2

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10;

Accepted Answer

The sequence starts with $a_1 = 1$. Using the recursion formula $a_{n+1} = \frac{a_n}{n+2}$, we calculate the next terms as follows: $a_2 = \frac{1}{3}$, $a_3 = \frac{1/3}{4} = \frac{1}{12}$, $a_4 = \frac{1/12}{5} = \frac{1}{60}$, and $a_5 = \frac{1/60}{6} = \frac{1}{360}$.

Question 3

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10; &#10;A) 1, -2, 3, -4, 5&#10;B) 1, -1, 2, -3, 5&#10;C) 1, 2, 1, 0, -1&#10;D) 1, 2, 1, -1, -2

Accepted Answer

Using the recursion formula, we have:
$$a_3 = a_2 - a_1 = 2-1 = 1$$
$$a_4 = a_3 - a_2 = 1-2 = -1$$
$$a_5 = a_4 - a_3 = -1-1 = -2$$
Thus, the first five terms of the sequence are 1, 2, 1, -1, -2. Therefore, the answer is choice D.

Question 4

Write the first four elements of the sequence.&#10;

Accepted Answer

The answer of Write the first four elements of the...

Question 5

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10; &#10;A) 1, 1, 1, 1, 1&#10;B) 1, 2, 4, 8, 16, 32&#10;C) 1, 2, 4, 8, 16&#10;D) 1, 3, 5, 7, 9

Accepted Answer

The sequence starts with $a_1 = 1$, and each subsequent term is the square of the previous term. Thus, all terms remain 1: $1, 1, 1, 1, 1$.

Question 6

Find a formula for the nth term of the sequence.&#10;5, 6, 7, 8, 9 (integers beginning with 5)

Accepted Answer

The sequence starts at 5 and increases by 1 for each subsequent term. To find the nth term, we start with n and add 4 to get to the starting point of 5 when n=1. Therefore, the formula is $a_n = n + 4$.

Question 7

Write the first four elements of the sequence.&#10;

Accepted Answer

Plug in $n=1,2,3,4$ and simplify to get the first four terms of the sequence: $2,\frac{9}{4},\frac{64}{27},\frac{625}{256}$. These are the same as those listed in choice A.

Question 8

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10; &#10;A) 2, -2, -2, 2, 2&#10;B) 2, -2, 2, -2, 2&#10;C) -2, 2, -2, 2, -2&#10;D) 2, 2, -2, -2, 2

Accepted Answer

The first term is 2. The second term is (-1)^1 * 2 = -2. The third term is (-1)^2 * (-2) = 2. The fourth term is (-1)^3 * 2 = -2. The fifth term is (-1)^4 * (-2) = 2.

Question 9

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10; &#10;A) 1, 1, 2, 3, 5&#10;B) 1, 5, 6, 7, 8&#10;C) 1, 5, 6, 12, 18&#10;D) 1, 5, 6, 11, 17

Accepted Answer

Using the given recursion formula, we have $$a_3=a_2+a_1=5+1=6$$ $$a_4=a_3+a_2=6+5=11$$ $$a_5=a_4+a_3=11+6=17$$ Therefore, the first five terms of the sequence are 1, 5, 6, 11, 17, and the best choice is $\textbf{(D)}$.

Question 10

Write the first four elements of the sequence.&#10; &#10;A) 0, 0, 0, 0&#10;B) 1, 1, 1, 1&#10;C) 0, 1, 0, -1&#10;D) 1, 0, -1, 0

Accepted Answer

The sine of any integer multiple of $\pi$ is 0, so for $n = 1, 2, 3, 4$, $\sin(n\pi) = 0$ for all these values.

Question 11

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10;

Accepted Answer

The answer of A recursion formula and the initial term(s)...

Question 12

Find a formula for the nth term of the sequence.&#10;-5, -4, -3, -2, -1 (integers beginning with -5)

Accepted Answer

The answer of Find a formula for the nth term...

Question 13

Write the first four elements of the sequence.&#10;

Accepted Answer

The answer of Write the first four elements of the...

Question 14

Find a formula for the nth term of the sequence.&#10;

Accepted Answer

The answer of Find a formula for the nth term...

Question 15

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10; &#10;A) 1, 5, 25, 125, 625&#10;B) 1, 6, 11, 16, 21&#10;C) 5, 6, 7, 8, 9&#10;D) 5, 25, 125, 625, 3125

Accepted Answer

The answer of A recursion formula and the initial term(s)...

Question 16

Find a formula for the nth term of the sequence.&#10;, , , 10, 12 (every other integer starting with )

Accepted Answer

The answer of Find a formula for the nth term...

Question 17

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10;

Accepted Answer

The answer of A recursion formula and the initial term(s)...

Question 18

Find a formula for the nth term of the sequence.&#10;6, -6, 6, -6, 6 (6's with alternating signs)

Accepted Answer

The answer of Find a formula for the nth term...

Question 19

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.&#10; &#10;A) 6, 11, 16, 21, 26&#10;B) 1, 6, 11, 16, 21&#10;C) 1, 6, 11, 16, 21, 26&#10;D) 1, 5, 25, 125, 625, 3125

Accepted Answer

The answer of A recursion formula and the initial term(s)...

Question 20

Write the first four elements of the sequence.&#10;

Accepted Answer

The answer of Write the first four elements of the...

Question 21

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 22

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 23

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 24

Find a formula for the nth term of the sequence.&#10;

Accepted Answer

The answer of Find a formula for the nth term...

Question 25

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 26

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 27

Find a formula for the nth term of the sequence.&#10;0, -1, 0, 1, 0, -1, 0, 1 (0, -1, 0, 1 repeated)

Accepted Answer

The answer of Find a formula for the nth term...

Question 28

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10; &#10;A) 1&#10;B) ln 9&#10;C) 0&#10;D) Diverges

Accepted Answer

The answer of Find the limit of the sequence if...

Question 29

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10; &#10;A) 0&#10;B) 3&#10;C) 1&#10;D) Diverges

Accepted Answer

The answer of Find the limit of the sequence if...

Question 30

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 31

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10; &#10;A) ln 2&#10;B) 1&#10;C) 0&#10;D) Diverges

Accepted Answer

The answer of Find the limit of the sequence if...

Question 32

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 33

Find a formula for the nth term of the sequence.&#10;0, 0, 2, 2, 0, 0, 2, 2 (alternating 0's and 2's in pairs)

Accepted Answer

The answer of Find a formula for the nth term...

Question 34

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10; &#10;A) 0&#10;B) ln 1&#10;C) 1&#10;D) Diverges

Accepted Answer

The answer of Find the limit of the sequence if...

Question 35

Find a formula for the nth term of the sequence.&#10;0, 2, 2, 2, 0, 2, 2, 2 (0, 2, 2, 2 repeated)

Accepted Answer

The answer of Find a formula for the nth term...

Question 36

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 37

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 38

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10; &#10;A) 6&#10;B) 1&#10;C) 0&#10;D) Diverges

Accepted Answer

The answer of Find the limit of the sequence if...

Question 39

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10; &#10;A) 5&#10;B) 4&#10;C) 4.3&#10;D) Diverges

Accepted Answer

The answer of Find the limit of the sequence if...

Question 40

Find a formula for the nth term of the sequence.&#10;0, 2, 0, 2, 0 (alternating 0's and 2's)

Accepted Answer

The answer of Find a formula for the nth term...

Question 41

Determine if the sequence is monotonic and if it is bounded.&#10; &#10;A) not monotonic; unbounded&#10;B) nondecreasing; bounded&#10;C) decreasing; bounded&#10;D) increasing; unbounded

Accepted Answer

The answer of Determine if the sequence is monotonic and...

Question 42

Assume that the sequence converges and find its limit.&#10; &#10;A) -3&#10;B) -9&#10;C) 4&#10;D) 8

Accepted Answer

The answer of Assume that the sequence converges and find...

Question 43

Determine if the sequence is monotonic and if it is bounded.&#10; &#10;A) nonincreasing; unbounded&#10;B) not monotonic; unbounded&#10;C) nondecreasing; bounded&#10;D) nondecreasing; unbounded

Accepted Answer

The answer of Determine if the sequence is monotonic and...

Question 44

Determine if the sequence is monotonic and if it is bounded.&#10; &#10;A) nondecreasing; bounded&#10;B) decreasing; bounded&#10;C) increasing; unbounded&#10;D) not monotonic; bounded

Accepted Answer

The answer of Determine if the sequence is monotonic and...

Question 45

Determine whether the nonincreasing sequence converges or diverges.&#10; &#10;A) Converges&#10;B) Diverges

Accepted Answer

The answer of Determine whether the nonincreasing sequence converges or...

Question 46

Determine if the sequence is monotonic and if it is bounded.&#10; &#10;A) decreasing; unbounded&#10;B) not monotonic; bounded&#10;C) increasing; bounded&#10;D) nonincreasing; bounded

Accepted Answer

The answer of Determine if the sequence is monotonic and...

Question 47

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10; &#10;A) 1&#10;B) 5&#10;C) 0&#10;D) Diverges

Accepted Answer

The answer of Find the limit of the sequence if...

Question 48

Determine if the sequence is monotonic and if it is bounded.&#10; &#10;A) decreasing; unbounded&#10;B) nondecreasing; bounded&#10;C) not monotonic; unbounded&#10;D) nondecreasing; unbounded

Accepted Answer

The answer of Determine if the sequence is monotonic and...

Question 49

Determine if the sequence is monotonic and if it is bounded.&#10; &#10;A) not monotonic; bounded&#10;B) nondecreasing; bounded&#10;C) decreasing; bounded&#10;D) decreasing; unbounded

Accepted Answer

The answer of Determine if the sequence is monotonic and...

Question 50

Determine whether the nonincreasing sequence converges or diverges.&#10; &#10;A) Converges&#10;B) Diverges

Accepted Answer

The answer of Determine whether the nonincreasing sequence converges or...

Question 51

Determine whether the nonincreasing sequence converges or diverges.&#10; &#10;A) Converges&#10;B) Diverges

Accepted Answer

The answer of Determine whether the nonincreasing sequence converges or...

Question 52

Determine if the sequence is monotonic and if it is bounded.&#10; &#10;A) not monotonic; unbounded&#10;B) nondecreasing; bounded&#10;C) nondecreasing; unbounded&#10;D) increasing; bounded

Accepted Answer

The answer of Determine if the sequence is monotonic and...

Question 53

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 54

Assume that the sequence converges and find its limit.&#10; &#10;A) 5&#10;B) -5&#10;C) 1&#10;D) 25

Accepted Answer

The answer of Assume that the sequence converges and find...

Question 55

Determine if the sequence is monotonic and if it is bounded.&#10; &#10;A) nondecreasing; bounded&#10;B) not monotonic; bounded&#10;C) increasing; unbounded&#10;D) decreasing; bounded

Accepted Answer

The answer of Determine if the sequence is monotonic and...

Question 56

Determine whether the nonincreasing sequence converges or diverges.&#10; &#10;A) Diverges&#10;B) Converges

Accepted Answer

The answer of Determine whether the nonincreasing sequence converges or...

Question 57

Assume that the sequence converges and find its limit.&#10; &#10;A) 5&#10;B) 6&#10;C) 7&#10;D) 8

Accepted Answer

The answer of Assume that the sequence converges and find...

Question 58

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 59

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10; &#10;A) 6&#10;B) ln 6&#10;C) 0&#10;D) Diverges

Accepted Answer

The answer of Find the limit of the sequence if...

Question 60

Find the limit of the sequence if it converges; otherwise indicate divergence.&#10;

Accepted Answer

The answer of Find the limit of the sequence if...

Question 61

Find the smallest value of N that will make the inequality hold for all n > N.&#10; &#10;A) 16&#10;B) 15&#10;C) 14&#10;D) 13

Accepted Answer

The answer of Find the smallest value of N that...

Question 62

&#10; &#10;A) L = ln 2, N = 6&#10;B) L = 0, N = 7&#10;C) L = ln 4, N = 8&#10;D) diverges

Accepted Answer

The answer of  &#10; &#10;A) L = ln 2,...

Question 63

&#10; &#10;A) L = 1, N = 28&#10;B) L = 1, N = 6&#10;C) L = 0, N = 28&#10;D) diverges

Accepted Answer

The answer of  &#10; &#10;A) L = 1, N...

Question 64

&#10; &#10;A) L = 1, N = 738&#10;B) L = 1, N = 1549&#10;C) L = 0, N = 1549&#10;D) diverges

Accepted Answer

The answer of  &#10; &#10;A) L = 1, N...

Deck 11: Infinite Sequences and Series