The problem involves a sequence defined recursively. From the image, we have the following details:

• The first term $a_1 = 2$,
• The recursive formula for the $n$-th term is $a_n = 2 \cdot a_{n-1} - 1$,
• The task is to find the 5th term of the sequence $a_5$.

Step-by-Step Solution:

1. Start with the initial condition: $a_1 = 2$

2. Calculate the subsequent terms using the recurrence relation:

   For $n = 2$: $a_2 = 2 \cdot a_1 - 1 = 2 \cdot 2 - 1 = 3$

   For $n = 3$: $a_3 = 2 \cdot a_2 - 1 = 2 \cdot 3 - 1 = 5$

   For $n = 4$: $a_4 = 2 \cdot a_3 - 1 = 2 \cdot 5 - 1 = 9$

   For $n = 5$: $a_5 = 2 \cdot a_4 - 1 = 2 \cdot 9 - 1 = 17$

3. Answer: The 5th term of the sequence is $a_5 = 17$.

Final Answer:

Option 4) 17

Do you want more details on recurrence relations or related examples?

5 Related Questions:

1. What is the general formula for a sequence defined recursively?
2. How do recurrence relations apply to solving sequences in mathematics?
3. What are other examples of sequences defined by similar recurrence relations?
4. How can we find the $n$-th term of a sequence without calculating all intermediate terms?
5. What is the role of recurrence relations in computer science and algorithms?

Tip:

When working with recursive sequences, always start by calculating a few terms manually to detect patterns or verify the calculations.