Answer and Step-by-step explanation:
The sequence of numbers vector{b} is defined recursively as follows:
- Base case: b_{0} = 3
- Recursive case: For n ≥ 1, we have b_{n} = 2 b_{n-1} + n
To better understand this recursive definition, let's examine how the sequence is generated step by step:
1. Starting with the base case, we have b_{0} = 3.
2. Using the recursive case, we can find the next term in the sequence:
- For n = 1: b_{1} = 2 * b_{0} + 1 = 2 * 3 + 1 = 7
- For n = 2: b_{2} = 2 * b_{1} + 2 = 2 * 7 + 2 = 16
- For n = 3: b_{3} = 2 * b_{2} + 3 = 2 * 16 + 3 = 35
- And so on...
We can continue this process to generate more terms in the sequence by substituting the previous term into the recursive formula.
In summary, the sequence vector{b} is defined recursively, starting with the base case b_{0} = 3, and each subsequent term is obtained by multiplying the previous term by 2 and adding the value of n.