5.3 MTH 288

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and f(n+1)=f(n)2f(n− 1)�(�+1)=�(�)2�(�⁢− 1) for n = 1, 2, ... f(2) = (You must provide an answer before moving to the next part.)

-4

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and f(n+1)=f(n)2f(n− 1)�(�+1)=�(�)2�(�⁢− 1) for n = 1, 2, ... f(4) = (You must provide an answer before moving to the next part.)

-4096

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and f(n + 1) = f(n) + 3f(n − 1) for n = 1, 2, .. f(5) =

17

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and f(n + 1) = f(n) + 3f(n − 1) for n = 1, 2, .. f(4) = (You must provide an answer before moving to the next part.)

2

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and f(n+1)=f(n)2f(n− 1)�(�+1)=�(�)2�(�⁢− 1) for n = 1, 2, ... f(3) = (You must provide an answer before moving to the next part.)

32

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and f(n + 1) = f(n) + 3f(n − 1) for n = 1, 2, .. f(3) = (You must provide an answer before moving to the next part.)

5

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and f(n+1)=f(n)2f(n− 1)�(�+1)=�(�)2�(�⁢− 1) for n = 1, 2, ... f(5) =

536,870,912

Let F be the function such that F(n) is the sum of the first n non-negative integers. Give a recursive definition of F(n).

F(0) = 0, F(n) = F(n - 1) + n for all n ≥ 1

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. Prove by mathematical induction that the formula found in the previous problem is valid. First, outline the proof by clicking and dragging to complete each statement.

Let P(n) be the proposition that: f(n) = 4-n basis step: p(1) states that f(1) = 4 -1 =3, which fits the definition of f inductive step: assume that: f(k) = 4-k for an arbitrary integer k >0 show that Ak(P(k) -> p(k + 1)) is true, that is, Ak(f(k) = 4 -k -> f(k+ 1)= 4 - (k+1)) we have completed the basis step and the inductive step. by mathematical induction, we know that: P(n) is true for all integers n>0 f((k+1)-1)-1 f(k) -1 (4-k)-1 4-(k+1)

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. f(0) = 0, f(n) = 2f(n - 2) for n ≥ 1 Choose the correct statement. (You must provide an answer before moving to the next part.)

The definition of f is not valid because defining f(1) would require f(-1), which is not available.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. f(0) = 1, f(1) = 2, f(n) = 2f(n - 2) for n ≥ 2 Choose the correct statement. (You must provide an answer before moving to the next part.)

The definition of f is valid.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. f(0) = 1, f(n) = 3f(n - 1) if n is odd and n ≥ 1 and f(n) = 9f(n - 2) if n is even and n ≥ 2 Choose the correct statement. (You must provide an answer before moving to the next part.)

The definition of f is valid.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. f(0) = 1, f(n) = f(n - 1) - 1 for n ≥ 1 Choose the correct statement. (You must provide an answer before moving to the next part.)

The definition of f is valid.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. f(0) = 2, f(1) = 3; f(n) = f(n - 1) - 1 for n ≥ 2 Choose the correct statement. (You must provide an answer before moving to the next part.)

The definition of f is valid.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Give a recursive definition of the sequence {an}, n = 1, 2, 3,... if an = n(n + 1)

a1 = 2, an = an - 1 + 2n, n > 1

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. Given the function f(n) defined as f(0) = 1, f(n) = f(n - 1) - 1 for n ≥ 1. Choose the correct formula for f(n) when n is a nonnegative integer. (You must provide an answer before moving to the next part.)

f(n) = 1 - n

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. Prove by mathematical induction that the formula determined in the previous problem is valid. First, outline the proof by clicking and dragging to complete each statement.

f(n) = 2^ ([n+1/2]) f(0) = 2^[(n+1)/2] = 1 and f(1) = 2^ [1+1/2] = 2. each of which fits the definition of f f(j) = 2^ [ j+1/2] for all j = 1.2,....,k, where k >0 is an arbitrary integer Ak(p(1) ^ p(2) ^...^p(k) -> p(k+1)) is true, that is, Ak(f(j) = 2 ^[j+1/2] for j = 1,2....k-> f(k+1) = 2^[(k+1) +1/2]) p(n) is true for all integers n>_0 2f((k+1)-2) 2f(k-1) 2* 2 [(k-1)+1/2] 2^[(k/2)]+1 2^ [(k+1)+1/2]

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. Choose the correct formula for f(n) when n is a nonnegative integer. (You must provide an answer before moving to the next part.)

f(n) = 2⌊n+12⌋2⌊�+12⌋ for n ≥ 0

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. Prove by mathematical induction that the formula found in the previous problem is valid. First, outline the proof by clicking and dragging to complete each statement.

f(n) = 3^n f(0) = 3^0 = 1 and f(1) = 3^1 = 3, each of which fits the definition of f f(j) = 3^j for all j = 1,2....k where k > 0 is an arbitrary integer Ak(p(1) ^ p(2) ^..^p(k) - > p(k+1)) is true, that is, Ak (f(j) = 3^j for j = 1,2...k -> f(k+1) = 3^k+1 p(n) is true for all integers n>0 3f((k+1) -1) if k is off and 9f((k+1) -2) if k is even 3f(k) if k is odd and 9f(k-1) if k is even 3*3^k if k is odd and 9*3^k-1 if k is even 3^k+1 whether k is odd or even

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. The definition of f is valid; and f(0) = 1, f(n) = 3f(n - 1) if n is odd and n ≥ 1 and f(n) = 9f(n - 2) if n is even and n ≥ 2. Choose the correct formula for f(n) when n is a nonnegative integer. (You must provide an answer before moving to the next part.)

f(n) = 3^n for n ≥ 0

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and f(n + 1) = f(n) + 3f(n − 1) for n = 1, 2, .. f(2) = (You must provide an answer before moving to the next part.)

-1

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. Prove by mathematical induction that the formula found in the previous problem is valid. First, outline the proof by clicking and dragging to complete each statement.

Let P(n) be the proposition that: f(n) = 1 - n Basis step: p(0) states that: f(0) = 1 - 0 = 1, which is the value given for f(0) in the definition of f inductive step: assume that: f(k) = 1 - k for an arbitrary integer k >_ 0 show that: Ak(P(k) -> P (k+1)) is true , that is Ak (f(k) = 1 - k -> f(k+1) = 1 - (k + 1)) we have completed the basis step and the inductive step. by mathematical induction, we know that: p(n) is true for all integers n >_ 0 Second, click and drag expressions to fill in the details of showing that ∀∀ k(P(k) → P(k + 1)) is true, thereby completing the induction step. (You must provide an answer before moving to the next part.) f((k+1)-1)-1 f(k)-1 (1-k)-1 1-(k+1)

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Give a recursive definition of the sequence {an}, n = 1, 2, 3,... if an = 1 + (-1)n

a1 = 0, a2 = 2, an = an - 2 for all n ≥ 3

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Give a recursive definition of the sequence {an}, n = 1, 2, 3,... if an = 4n - 2

a1 = 2 and an + 1 = an + 4 for all n ≥ 1

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Give a recursive definition of the sequence {an}, n = 1, 2, 3,... if an = n2

a1= 1, an = an - 1 + 2n - 1, for all n > 1

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. Choose the correct formula for f(n) when n is a nonnegative integer. (You must provide an answer before moving to the next part.)

f(0) = 2; f(n) = 4 - n for n > 0