Fibonacci Using Recursion in DSA Go - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the base case in a recursive Fibonacci function?

The base case returns the first two Fibonacci numbers directly, usually when n is 0 or 1, to stop further recursion.

beginner

Explain how recursion works in calculating Fibonacci numbers.

Recursion calculates Fibonacci by calling itself with smaller inputs (n-1 and n-2) and combining their results until reaching base cases.

intermediate

What is the time complexity of the naive recursive Fibonacci algorithm?

The time complexity is exponential, O(2^n), because it recalculates the same Fibonacci numbers multiple times.

beginner

Show the first 5 Fibonacci numbers calculated by recursion starting from 0.

The first 5 Fibonacci numbers are: 0, 1, 1, 2, 3.

intermediate

Why is memoization helpful in recursive Fibonacci?

Memoization stores results of previous calls to avoid repeated calculations, improving efficiency from exponential to linear time.

What does the Fibonacci recursive function return for input n=0?

A n

B 1

C Undefined

D 0

By definition, Fibonacci(0) returns 0 as the first number in the sequence.

Which two previous Fibonacci numbers does the recursive function call to calculate Fibonacci(n)?

A Fibonacci(n-1) and Fibonacci(n-2)

B Fibonacci(n-2) and Fibonacci(n-3)

C Fibonacci(n-1) and Fibonacci(n-3)

D Fibonacci(n) and Fibonacci(n-1)

The recursive formula uses Fibonacci(n-1) + Fibonacci(n-2) to calculate Fibonacci(n).

What is the main drawback of the naive recursive Fibonacci implementation?

A Uses too much memory

B Exponential time complexity due to repeated calculations

C Does not return correct results

D Cannot handle base cases

Naive recursion recalculates many Fibonacci numbers multiple times, causing exponential time complexity.

Which technique can optimize recursive Fibonacci to run in linear time?

A Memoization

B Sorting

C Loop unrolling

D Binary search

Memoization caches results of recursive calls to avoid repeated work, improving efficiency.

What is the output of Fibonacci(4) using recursion starting with Fibonacci(0)=0?

A 5

B 2

C 3

D 4

Fibonacci sequence: 0,1,1,2,3; Fibonacci(4) is 3.

Describe how a recursive function calculates the nth Fibonacci number.

Think about how the function calls itself with smaller numbers until it reaches the simplest cases.

Explain why naive recursion for Fibonacci is inefficient and how memoization improves it.

Consider how many times the same Fibonacci numbers are calculated without and with memoization.

Practice

1. What does the Fibonacci sequence represent when calculated using recursion?

easy

A. Each number is the product of the two preceding numbers

B. Each number is the difference of the two preceding numbers

C. Each number is the sum of the two preceding numbers

D. Each number is the square of the previous number

Solution

1. Step 1: Understand Fibonacci sequence definition

   The Fibonacci sequence is defined such that each number is the sum of the two numbers before it.

2. Step 2: Relate to recursion

   Recursion calls the function for the two previous numbers and adds their results.

3. Final Answer:

   Each number is the sum of the two preceding numbers -> Option C

4. Quick Check:

   Fibonacci = sum of two previous numbers [OK]

Hint: Remember Fibonacci adds two previous numbers [OK]

Common Mistakes:

• Thinking Fibonacci multiplies previous numbers
• Confusing difference with sum
• Assuming Fibonacci squares previous number

2. Which of the following is the correct Go function signature for a recursive Fibonacci function?

easy

A. func fibonacci(n int) int

B. func fibonacci(n string) string

C. func fibonacci(n float64) float64

D. func fibonacci() int

Solution

1. Step 1: Identify parameter and return types

   The Fibonacci function takes an integer input and returns an integer output.

2. Step 2: Match correct Go syntax

   func fibonacci(n int) int matches the correct signature: func fibonacci(n int) int.

3. Final Answer:

   func fibonacci(n int) int -> Option A

4. Quick Check:

   Input and output are integers [OK]

Hint: Fibonacci uses int input and int output [OK]

Common Mistakes:

• Using string or float types incorrectly
• Omitting function parameters
• Wrong return type

3. What is the output of the following Go code?

func fibonacci(n int) int {
    if n <= 1 {
        return n
    }
    return fibonacci(n-1) + fibonacci(n-2)
}

func main() {
    fmt.Println(fibonacci(5))
}

medium

A. 5

B. 8

C. 3

D. 13

Solution

1. Step 1: Trace fibonacci(5) calls

   fibonacci(5) = fibonacci(4) + fibonacci(3), fibonacci(4) = 3, fibonacci(3) = 2, so total = 5.

2. Step 2: Calculate final value

   Sum of fibonacci(4) and fibonacci(3) is 3 + 2 = 5.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   Fibonacci(5) = 5 [OK]

Hint: Recall Fibonacci(5) = 5 from sequence 0,1,1,2,3,5 [OK]

Common Mistakes:

• Confusing index starting at 0 or 1
• Adding wrong recursive results
• Miscounting base cases

4. Identify the error in this recursive Fibonacci function and select the fix:

func fibonacci(n int) int {
    if n == 0 {
        return 0
    } else if n == 1 {
        return 1
    }
    return fibonacci(n-1) - fibonacci(n-2)
}

medium

A. Add a loop instead of recursion

B. Change base case n == 0 to n == 2

C. Remove else if condition

D. Change '-' to '+' in the return statement

Solution

1. Step 1: Check recursive return operation

   The Fibonacci sequence sums previous two numbers, but code subtracts them.

2. Step 2: Correct the operation

   Replace '-' with '+' to correctly add fibonacci(n-1) and fibonacci(n-2).

3. Final Answer:

   Change '-' to '+' in the return statement -> Option D

4. Quick Check:

   Fibonacci sums previous two numbers [OK]

Hint: Fibonacci adds, never subtracts previous two numbers [OK]

Common Mistakes:

• Using subtraction instead of addition
• Changing base cases incorrectly
• Removing necessary conditions

5. How can you optimize the recursive Fibonacci function to avoid repeated calculations?

hard

A. Increase the recursion depth limit

B. Use memoization to store and reuse results

C. Use a global variable to count calls

D. Replace recursion with a random number generator

Solution

1. Step 1: Understand repeated calculations problem

   Recursive Fibonacci recalculates same values many times, causing inefficiency.

2. Step 2: Apply memoization technique

   Memoization stores results of previous calls to reuse them, reducing repeated work.

3. Final Answer:

   Use memoization to store and reuse results -> Option B

4. Quick Check:

   Memoization optimizes recursion by caching [OK]

Hint: Cache results to avoid repeated recursion calls [OK]

Common Mistakes:

• Increasing recursion depth does not fix inefficiency
• Counting calls doesn't optimize calculation
• Random numbers unrelated to Fibonacci