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

Question

How could the code be altered to print the sum of the first 10 numbers in the sequence 3n -1?

Answer 1

To modify a recursive algorithm that sums the sequence of numbers from 1 to 100 to instead sum the first 10 numbers in the sequence defined by the formula \(3n - 1\), we can follow these steps:

1. Identify the sequence: The sequence defined by \(3n - 1\) for \(n = 1, 2, \ldots, 10\) is:

   • For \(n=1\): \(3 \cdot 1 - 1 = 2\)
   • For \(n=2\): \(3 \cdot 2 - 1 = 5\)
   • For \(n=3\): \(3 \cdot 3 - 1 = 8\)
   • For \(n=4\): \(3 \cdot 4 - 1 = 11\)
   • For \(n=5\): \(3 \cdot 5 - 1 = 14\)
   • For \(n=6\): \(3 \cdot 6 - 1 = 17\)
   • For \(n=7\): \(3 \cdot 7 - 1 = 20\)
   • For \(n=8\): \(3 \cdot 8 - 1 = 23\)
   • For \(n=9\): \(3 \cdot 9 - 1 = 26\)
   • For \(n=10\): \(3 \cdot 10 - 1 = 29\)

2. Calculate the sum recursively. The base case will occur when \(n\) reaches 0, and the recursive case will calculate the current term of the sequence from the formula and add it to the result of the recursive call for \(n-1\).

Here is how the code might look in Python:

def sum_sequence(n):
    # Base case: if n is 0, return 0 as there is nothing to sum
    if n == 0:
        return 0
    else:
        # Recursive case: calculate the current term (3*n - 1)
        current_term = 3 * n - 1
        # Recursive call with n-1
        return current_term + sum_sequence(n - 1)

# We want the sum of the first 10 terms in the sequence 3n - 1
result = sum_sequence(10)
print(f"The sum of the first 10 numbers in the sequence 3n - 1 is: {result}")

Explanation of the Code:

• The function sum_sequence(n) takes an integer n representing how many terms to sum.
• When n is 0, we return 0, which is the base case of our recursion.
• For any other value of n, we calculate the current term using the formula \(3n - 1\) and add it to the recursive call to sum_sequence(n - 1).
• We call sum_sequence(10) to sum the first 10 terms of the sequence and print the result.

This code snippet will compute the sum of the first 10 numbers in the sequence \(3n - 1\) correctly.