2.4 MTH 288

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Compute each of these double sums. ∑3i= 1∑2j= 1(i− j)∑�⁢= 13∑�⁢= 12(�⁢− �) = _____

3

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the solution to the recurrence relation by using an iterative approach. Use the following procedure for the recurrence relation an = 2nan - 1 with the initial condition a0 = 3. an = 2nan - 1 = 2n(2(n - 1)an - 2) = 22(n(n - 1))an - 2 = 22(n(n - 1))(2(n - 2)an - 3) = 23(n(n - 1)(n - 2))an - 3 = ⋅ ⋅ ⋅ continuing in the same manner = 2nn(n - 1)(n - 2)(n - 3) ⋅ ⋅ ⋅ (n - (n - 1))an - n = 2nn(n - 1)(n - 2)(n - 3) ⋅ ⋅ ⋅ 1 ⋅ a0 = _____

3 · 2n n!

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Compute each of these double sums. ∑2i= 0∑3j= 0i2j3∑�⁢= 02∑�⁢= 03�2�3 = _____

180

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence that begins with 2 and in which each successive term is 3 more than the preceding term. The first 10 terms are

2,5,8,11,14,17,20,23,26,29

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the solution to the recurrence relation by using an iterative approach. The recurrence relation an = -an - 1 + n - 1 with the initial condition a0 = 7 an = n − 1 − an − 1= n − 1 − ((n − 1 − 1) − an − 2= (n − 1) − (n − 2) + an − 2= (n − 1) − (n − 2) + ((n − 2 − 1) − an − 3)= (n − 1) − (n − 2) + (n − 3) − an − 3= ⋅ ⋅ ⋅ continuing in the same manner= (n − 1) − (n − 2) + · · · + (−1)n − 1(n − n) + (−1)nan − n= _____

(2n− 1+ (−1)n/4) + (−1)n⋅ 7

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence whose nth term is n! − 2n where n ≥ 1. The first 10 terms are

-1,-2,-2,8,88,656,4912,40064,362368, 3627776

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence that lists each positive integer three times, in an increasing order. The first 10 terms are

1,1,1,2,2,2,3,3,3,4

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence that lists the odd positive integers in an increasing order, listing each odd integer twice. The first 10 terms are

1,1,3,3,5,5,7,7,9,9

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence whose nth term is the number of bits in the binary expansion of the number n. (Note: The value of n starts from 1) The first 10 terms are

1,2,2,3,3,3,3,4,4,4

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the solution to the recurrence relation by using an iterative approach. Arrange the following steps of the recurrence relation an = an - 1 - n with the initial condition a0 = 4.

1. -n + (-(n - 1) + an - 2) = -(n + (n - 1)) + an - 2 2. -(n + (n - 1)) + (-(n - 2) + an - 3) = -(n + (n - 1) + (n - 2)) + an - 3 3. -(n + (n - 1) + (n - 2) + ⋅ ⋅ ⋅ + (n - (n - 1))) + an - n 4. -(n + (n - 1) + (n - 2) + ⋅ ⋅ ⋅ + 1) + a0 5. (−n(n+ 1))/2+ 4= (−n2− n + 8)/2

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence where the nth term is the number of letters in the English word for the number n. (Note: The value of n starts from 1.) The first 10 terms are

3,3,5,4,4,3,5,5,4,3

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence that begins with 3, where each succeeding term is twice the preceding term. The first 10 terms are

3,6,12,24,48,96,192,384,768,1536

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Compute each of these double sums. ∑3i = 0∑2j = 0(3i+2j)∑� = 03∑� = 02(3�+2�) = _____

78

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Compute each of these double sums. ∑3i= 1∑2j= 0j∑�⁢= 13∑�⁢= 02� = _____

9

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the solution to the recurrence relation by using an iterative approach. Identify the missing steps, denoted by A and B, for the recurrence relation an = (n + 1)an - 1 with the initial condition a0 = 2. (Check all that apply.) an = (n + 1)an − 1 = (n + 1)nan − 2= (n + 1)n(n − 1)an − 3 = (n + 1)n(n − 1)(n − 2)an − 4 = ⋅ ⋅ ⋅ continuing in the same manner = (n + 1)n(n − 1)(n − 2)(n − 3) · · · (n − (n − 2)) an − n= A= B

A = (n + 1)n(n − 1)(n − 2)(n − 3) · · · 2 · a0 B = 2(n + 1)!

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. What are the terms a0, a1, a2, and a3 of the sequence {an}? Find the first four terms of the sequence given below. The values of the first four terms of the sequence are a0 =

Plug n = 0,1, 2, 3 into the formula: . a0 = 0 a1 = 1 a2 = 2 a3 = 3

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these lists of integers, find a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, . . .

The next three terms are 0, 0, 0.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these lists of integers, find a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, . . .

The next three terms are 1100, 1101, 1110.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these lists of integers, find a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, . . .

The next three terms are 123, 146, 171.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these lists of integers, find a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 2, 4, 16, . . .

The next three terms are 256, 65536, 4294967296.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these lists of integers, find a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, . . .

The next three terms are 47, 51, 55.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these lists of integers, find a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, . . .

The next three terms are 59048, 177146, 531440.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these lists of integers, find a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, . . .

The next three terms are 654729075, 13749310575, 316234143225.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. For each of these lists of integers, find a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 1, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, . . .

The next three terms are 8, 8, 8.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the solution to the recurrence relation by using an iterative approach. The recurrence relation an = -an - 1 with the initial condition a0 = 5

The solution for the recurrence relation is an = −an − 1 = (−1)2an−2 = ⋅ ⋅ ⋅ = (−1)nan − n = (−1)na0 = 5 ⋅ (−1)n

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the solution to the recurrence relation by using an iterative approach. The recurrence relation an = an - 1 + 3 with the initial condition a0 = 1

The solution for the recurrence relation is an = 3 + an − 1 = 3 + 3 + an − 2 = 2 · 3 + an − 2 = 3 · 3 + an − 3 = ⋅ ⋅ ⋅= n · 3 + an − n = n · 3 + a0 = 3n + 1

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. For the sequence an = nan - 1 + a2n - 2, a0 = −1 and a1 = 0, the values of the first six terms are

a0 = -1 a1 = 0 a2 = 1 a3 = 3 a4 = 13 a5 = 74

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. For the sequence an = −2an − 1 and a0 = −1, the values of the first six terms are

a0 = -1 a1 = 2 a2 = -4 a3 = 8 a4 = -16 a5 = 32

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. For the sequence an = an - 1 − an - 2 + an - 3, a0 = 1, a1 = 1, and a2 = 2, the values of the first six terms are

a0 = 1 a1 = 1 a2 = 2 a3 = 2 a4 = 1 a5 = 1

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. For the sequence an = 3a2n - 1 and a0 = 1, the values of the first six terms are

a0 = 1 a1 = 3 a2 = 27 a3 = 2187 a4 = 14348907 a5 = 617673396283947

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. For the sequence an = an - 1 - an - 2, a0 = 2, and a1 = -1, the values of the first six terms are

a0 = 2 a1 = -1 a2 = -3 a3 = -2 a4 = 1 a5 = 3

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence whose first term is 2, second term is 4, and each succeeding term is the sum of the two preceding terms. The first 10 terms are

a0 = 2 and a1 = 4 Plug n = 2, 3, . . ., 9 into the formula: an = an - 1 + an - 2. 2,4,6,10,16,26,42,68,110,178

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. What are the terms a0, a1, a2, and a3 of the sequence {an}? For the sequence an = 2n + 1, identify the values of a0 = _____, a1 = _____, a2 = _____, and a3 = _____.

a0 = 2, a1 = 3, a2 = 5, and a3 = 9

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. What are the terms a0, a1, a2, and a3 of the sequence {an}? For the sequence an = ⌊n/2⌋⌊�/2⌋ , the values of the first four terms of the sequence are a0 =

a0=0 a1=0 a2=1 a3=1

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. What are the terms a0, a1, a2, and a3 of the sequence {an}? For the sequence an = (n + 1)(n + 1), the values of the first four terms of the sequence are a0 =

a0=1 a1=4 a2=27 a3=256

The sequence {an} is a solution of the recurrence relation an = 8an − 1 − 16an − 2 if _____. (Check all that apply.)

an = 0 an = 4n an = n4n an = 2 · 4n + 3n4n

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find the solution to the recurrence relation by using an iterative approach. Identify the steps for the recurrence relation an = 2an - 1 - 3 with the initial condition a0 = -1. (Check all that apply.)

an = −3 + 2(−3) + 4an − 2 an = −3 + 2(−3) + 4(−3) + 8an − 3 an = −3 + 2(−3) + 4(−3) + 8(−3) + 16an − 4 Continuing the same manner, an = −3(1 + 2 + 4 + · · · + 2n − 1) + 2nan − n = −3(2n − 1) + 2n(−1) = −2n + 2 + 3