College

C++

You have the following sequence of numbers:
1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, ...

Find the recursive rule from the above series, give a recursive definition, and write a function that implements the definition.

(Note: \( f(1) = 1 \), \( f(2) = 2 \), \( f(3) = 4 \), \( f(4) = 7 \), ...)

Answer :

Final answer:

The recursive rule for the given sequence is f(n) = f(n-1) + n-1, where n is the position of the term in the sequence. Here is a C++ function that implements this recursive rule:

Explanation:

To find the recursive rule for the given sequence, we can observe that each term is obtained by adding the position of the term to the previous term. Let's break down the sequence:

  1. The first term is 1.
  2. The second term is obtained by adding 1 to the previous term: 1 + 1 = 2.
  3. The third term is obtained by adding 2 to the previous term: 2 + 2 = 4.
  4. The fourth term is obtained by adding 3 to the previous term: 4 + 3 = 7.
  5. And so on.

Based on this pattern, we can define the recursive rule as follows:

f(n) = f(n-1) + n-1

where n is the position of the term in the sequence.

To implement this recursive rule in a function, we can use the following C++ code:

This function takes an integer n as input and returns the nth term of the sequence using the recursive rule. The base case is when n is equal to 1, in which case the function returns 1. Otherwise, the function calls itself with n-1 as the argument and adds n-1 to the result.

Learn more about recursive rule and function implementation for a number sequence here:

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