Let's answer each question step-by-step as you requested, with clear explanations and a final summary at the end. --- ## **4. Euclidean Algorithm for gcd(a, b)** ### **Step 1: Introduction** The Euclidean algorithm is a method for finding the greatest common divisor (gcd) of two integers. There are both iterative and recursive versions. ### **Step 2: Iterative Version** Given two positive integers \( a > b \): - While \( b \neq 0 \): - Set \( r = a \mod b \) - Set \( a = b \), \( b = r \) - Return \( a \) as the gcd. ### **Step 3: Recursive Version** Given two positive integers \( a > b \): - If \( b = 0 \), return \( a \) - Otherwise, return gcd(\( b, a \mod b \)) --- ## **5. Recursive Selection Sort** ### **Step 1: Introduction** Selection sort finds the minimum element and places it at the start, then recursively sorts the rest. ### **Step 2: Recursive Algorithm** Given a list of 10 unsorted integers: - Find the minimum value in the list. - Place it at the first position. - Recursively sort the remaining 9 elements. - Base case: If the list length is 1, return. --- ## **6. Binary Search Algorithm** ### **Step 1: Introduction** Binary search efficiently checks if a value \( x \) exists in a **sorted** list. ### **Step 2: Algorithm** Given a sorted list of 10 integers and value \( x \): - Compare \( x \) to the middle element. - If equal, return "found". - If \( x \) is less, search the left half recursively. - If \( x \) is more, search the right half recursively. - If the sublist is empty, return "not found". --- ## **7. String Pattern Search** ### **Step 1: Introduction** Pattern search finds the location of a substring (pattern) within a text string. ### **Step 2: Algorithm (Naive)** Given a text string and a pattern string: - For every possible starting position in the text: - Check if the substring matches the pattern. - If yes, return the starting position. - If not found, return a message saying not found. --- ## **8. Golden Ratio and Fibonacci Sequence** ### **Step 1: Introduction** The ratio of consecutive Fibonacci numbers approaches the golden ratio as \( n \) increases. ### **Step 2: Calculation Steps** - Compute Fibonacci numbers up to \( n = 10, 15, 25, 50, 100 \). - Calculate \( F(n+1)/F(n) \) for each \( n \). - Compare each ratio to \( \frac{1+\sqrt{5}}{2} \approx 1.618 \). --- ## **9. Sequential vs Binary Search Work** ### **Step 1: Introduction** Sequential search checks each entry, binary search repeatedly halves the list. ### **Step 2: Work Calculation** - For sequential search: Number of comparisons is \( n \). - For binary search: Number of comparisons is \( 1 + \log_2 n \). - Compute and plot both for \( n = 1 \) to \( 100 \). --- ## **Final Answer (Concise Summary)** Each problem involves implementing or analyzing classic algorithms: Euclidean algorithms for gcd, recursive selection sort, binary search, and string pattern search. We also explored the connection between Fibonacci ratios and the golden ratio, and compared the work (number of comparisons) in sequential and binary search, plotting their growth as \( n \) increases. **What we did in each step:** We introduced each algorithm or concept, explained its step-by-step execution, and provided detailed methods for solving or analyzing the problem, finishing with a concise summary for clarity.