Question

Question

Given the code for the recursive algorithm that sums the sequence of numbers from one to one hundred:

def sum(n):
if n < 1:
return 0
else:
return n + sum(n - 1)
result = sum(100)
print(result)
How could the code be altered to print the sum of the first 6 terms in the sequence, 7n+1 ?

(1 point)
Responses

def sum(n):
if n <= 0:
return 0
else:
return n + sum(7 * n - 6)
result = sum(6)
print(result)
def sum(n): if n <= 0: return 0 else: return n + sum(7 * n - 6) result = sum(6) print(result)

def sum(n):
if n == 0:
return 0
else:
return 7 * n + 1 + sum(n - 1)
result = sum(6)
print(result)
def sum(n): if n == 0: return 0 else: return 7 * n + 1 + sum(n - 1) result = sum(6) print(result)

def sum(n):
if n == 0:
return 0
else:
return 7 * n + 1 + sum(n - 1)
result = sum(100)
print(result)
def sum(n): if n == 0: return 0 else: return 7 * n + 1 + sum(n - 1) result = sum(100) print(result)

def sum(n):
if n == 0:
return 0
else:
return 7 * n + 1 + sum(n - 1)
result = sum(10)
print(result)
def sum(n): if n == 0: return 0 else: return 7 * n + 1 + sum(n - 1) result = sum(10) print(result)
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Answer 1

To print the sum of the first 6 terms of the sequence defined by the formula \( 7n + 1 \), we need to modify the recursive function accordingly. The correct function initializes with \( n = 6 \) and sums the values produced by the formula in each recursive call.

The correct option for the implementation is:

def sum(n):
    if n == 0:
       return 0
    else:
       return 7 * n + 1 + sum(n - 1)

result = sum(6)
print(result)

This code will correctly compute the sum of the sequence for the first 6 terms, \( 7(1) + 1, 7(2) + 1, 7(3) + 1, 7(4) + 1, 7(5) + 1, \) and \( 7(6) + 1 \). Each recursive call adds the value \( 7n + 1 \) for the current \( n \) and then calls itself with \( n-1 \).

Thus, the response structure looks like this:

def sum(n):
    if n == 0:
       return 0
    else:
       return 7 * n + 1 + sum(n - 1)
result = sum(6)
print(result)

This follows the pattern of computing the sum recursively while adhering to the structure of the original function you provided.

Answer 2

Question
What is the output of the greatest common divisor function gcd(27,12) ?

gcd(x,y)={x,gcd(y,x%y),y=0y>0 where x%y is remainder division or modulo

(1 point)
Responses

12
12

1
1

3
3

27
27
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Answer 3

To find the greatest common divisor (gcd) of 27 and 12 using the provided function \( \text{gcd}(x, y) \) defined as:

\[ \text{gcd}(x, y) = \begin{cases} x & \text{if } y = 0 \ \text{gcd}(y, x \mod y) & \text{if } y > 0 \end{cases} \]

we can apply the function step by step for \( \text{gcd}(27, 12) \).

1. Start with \( x = 27 \) and \( y = 12 \).

   • Since \( y \) is not zero, we compute \( \text{gcd}(12, 27 \mod 12) \).
   • Calculate \( 27 \mod 12 \): \[ 27 \div 12 = 2 \quad \text{(remainder 3)} \] So, \( 27 \mod 12 = 3 \).
   • Now we have \( \text{gcd}(12, 3) \).

2. Now we compute \( \text{gcd}(12, 3) \).

   • Since \( y \) is not zero, we compute \( \text{gcd}(3, 12 \mod 3) \).
   • Calculate \( 12 \mod 3 \): \[ 12 \div 3 = 4 \quad \text{(remainder 0)} \] So, \( 12 \mod 3 = 0 \).
   • Now we have \( \text{gcd}(3, 0) \).

3. Finally, \( \text{gcd}(3, 0) \) returns \( 3 \) because if \( y = 0 \), we return \( x \).

Thus, \( \text{gcd}(27, 12) = 3 \).

The output of the function \( \text{gcd}(27, 12) \) is 3.

So the correct response is:

3

Answer 4

Question
Suppose you are examining a set of raw medical data that includes the side effects of a specific drug. You are interested in finding out which side effects occur the most often. What would be an appropriate algorithm to find this data?(1 point)
Responses

binary search
binary search

linear search
linear search

for loop
for loop

bubble sort
bubble sort
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Answer 5

To determine which side effects of a specific drug occur most often in a set of raw medical data, you need to analyze the frequency of each side effect. The most appropriate approach for this task would be:

Linear search: This algorithm allows you to iterate through the data to count occurrences of each side effect. You can maintain a frequency count for each unique side effect and then determine which one has the highest frequency.

Here's a brief overview of why the other options are not suitable:

• Binary search: This algorithm works on sorted data and is used to find the position of a specific value, not to count occurrences.

• For loop: While a for loop can be used to iterate through the data, it is not an algorithm on its own and doesn't directly solve the problem of finding the most common side effects without additional logic for counting.

• Bubble sort: This algorithm is used for sorting data, not for counting occurrences, which is not what you need for this task.

Given this analysis, the correct response is:

linear search.

Answer 6

Question
Students are discussing variations of algorithms to find Fibonacci numbers, like fib(6). What is the best possible time complexity that they can get for this algorithm?(1 point)
Responses

O(n)
upper O left parenthesis n right parenthesis

O(1)
upper O left parenthesis 1 right parenthesis

O(n2)
upper O left parenthesis n squared right parenthesis

O(logn)
upper O left parenthesis log n right parenthesis
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Answer 7

The Fibonacci sequence can be computed using various algorithms, and the time complexity varies depending on the approach taken. Here are some common methods of computing Fibonacci numbers and their associated time complexities:

1. Recursive Method: This is the naive implementation where each Fibonacci number is computed recursively. The time complexity is \( O(2^n) \), which is exponential and not efficient.

2. Dynamic Programming (Memoization or Tabulation): By storing previously computed values, you can reduce the time complexity to \( O(n) \). This is achieved by either using an array to store results (tabulation) or storing results in a cache (memoization).

3. Iterative Approach: Similarly, using an iterative approach to calculate Fibonacci numbers in a loop can achieve \( O(n) \) time complexity.

4. Matrix Exponentiation: This advanced method can compute Fibonacci numbers in \( O(\log n) \) time, which is the most efficient in terms of time complexity.

5. Binet's Formula: Using a closed-form expression derived from Binet's formula can compute Fibonacci numbers in \( O(1) \) time, although it may involve floating point operations that can introduce precision issues for large \( n \).

Given these options, the best possible time complexity to find Fibonacci numbers, particularly with an efficient algorithm like matrix exponentiation, is:

O(log n).

So, the correct response is:

O(log n).