Question 1

What is a recursive formula?

Accepted Answer

A recursive formula is a formula that defines each term of a sequence using one or more previous terms. Instead of giving a direct way to find any term, it shows how to get from one term to the next. For example, in the sequence 2, 5, 8, 11..., the recursive formula is a₁ = 2, aₙ = aₙ₋₁ + 3, meaning each term equals the previous term plus 3. Recursive formulas are useful for understanding how sequences grow and for computing terms step by step. They're fundamental in computer science for recursive algorithms.

Question 2

What is the difference between recursive and explicit formulas?

Accepted Answer

Recursive formula defines each term using previous term(s): aₙ = aₙ₋₁ + 3. You must know earlier terms to find later ones. Good for understanding patterns and computer programming. Explicit formula gives any term directly: aₙ = 2 + 3(n-1). You can find any term without calculating previous ones. Good for finding specific terms quickly. Example: For sequence 2, 5, 8, 11... Recursive: a₁ = 2, aₙ = aₙ₋₁ + 3 (need previous). Explicit: aₙ = 2 + 3(n-1) (direct calculation). Both describe the same sequence differently.

Question 3

How do you write a recursive formula for an arithmetic sequence?

Accepted Answer

For an arithmetic sequence (constant difference between terms): Step 1: Identify the first term (a₁). Step 2: Find the common difference (d) by subtracting consecutive terms. Step 3: Write the recursive formula: a₁ = [first term], aₙ = aₙ₋₁ + d. Example: Sequence 3, 7, 11, 15... First term: a₁ = 3. Common difference: 7 - 3 = 4. Recursive formula: a₁ = 3, aₙ = aₙ₋₁ + 4. This means: add 4 to any term to get the next term. To find a₅: a₂ = 7, a₃ = 11, a₄ = 15, a₅ = 19.

Question 4

How do you write a recursive formula for a geometric sequence?

Accepted Answer

For a geometric sequence (constant ratio between terms): Step 1: Identify the first term (a₁). Step 2: Find the common ratio (r) by dividing consecutive terms. Step 3: Write the recursive formula: a₁ = [first term], aₙ = aₙ₋₁ × r. Example: Sequence 3, 6, 12, 24... First term: a₁ = 3. Common ratio: 6 ÷ 3 = 2. Recursive formula: a₁ = 3, aₙ = aₙ₋₁ × 2. This means: multiply any term by 2 to get the next term. To find a₅: a₂ = 6, a₃ = 12, a₄ = 24, a₅ = 48.

Question 5

What is the Fibonacci sequence recursive formula?

Accepted Answer

The Fibonacci sequence is a famous recursive sequence where each term is the sum of the two preceding terms. Recursive formula: F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂ (for n ≥ 3). This creates the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Each term is sum of previous two: 1+1=2, 1+2=3, 2+3=5, 3+5=8, etc. The Fibonacci sequence appears in nature (flower petals, spirals), mathematics (golden ratio), and computer science. It's purely recursive - there's no simple explicit formula, though Binet's formula provides a complex one using the golden ratio.

Question 6

How do you find the explicit formula from a recursive formula?

Accepted Answer

Converting recursive to explicit depends on sequence type: Arithmetic (aₙ = aₙ₋₁ + d): Explicit is aₙ = a₁ + (n-1)d. Count how many times d is added. Geometric (aₙ = aₙ₋₁ × r): Explicit is aₙ = a₁ × r^(n-1). Count how many times r is multiplied. Fibonacci (aₙ = aₙ₋₁ + aₙ₋₂): Very complex (Binet's formula involves golden ratio). General process: 1) Identify sequence type. 2) Find pattern in terms. 3) Express nth term directly. Example: Recursive a₁ = 5, aₙ = aₙ₋₁ + 2 → Terms: 5, 7, 9, 11... → Explicit: aₙ = 5 + 2(n-1) = 2n + 3.

Question 7

What are real-world applications of recursive formulas?

Accepted Answer

Recursive formulas appear throughout real life: Finance: Compound interest (new balance = old balance × (1+rate)), loan amortization. Biology: Population growth models (Pₙ = r × Pₙ₋₁), bacterial reproduction. Computer Science: Recursive algorithms (factorial, tree traversal), dynamic programming. Physics: Iterative calculations, numerical methods. Economics: Economic models, supply chain. Fractals: Each iteration uses previous result (Mandelbrot set). Games: Tower of Hanoi, Fibonacci nim. Understanding recursion helps in programming, modeling growth, and solving problems where current state depends on previous state.

Question 8

How do you solve for a specific term using recursive formula?

Accepted Answer

To find a specific term using recursive formula, calculate each term in sequence: Step 1: Start with given initial term(s). Step 2: Apply recursive rule to find next term. Step 3: Repeat until reaching desired term. Example: Find a₆ for a₁ = 3, aₙ = aₙ₋₁ + 4. Calculate: a₁ = 3, a₂ = 3 + 4 = 7, a₃ = 7 + 4 = 11, a₄ = 11 + 4 = 15, a₅ = 15 + 4 = 19, a₆ = 19 + 4 = 23. This sequential approach is necessary for recursive formulas. For large n, explicit formulas are more efficient. Computers handle recursive calculations well through recursion or iteration.

Question 9

What is the difference between arithmetic and geometric sequences?

Accepted Answer

Arithmetic sequences have a constant difference (d) added between terms: aₙ = aₙ₋₁ + d. Pattern: add same amount each time. Example: 2, 5, 8, 11, 14 (add 3). Linear growth. Explicit: aₙ = a₁ + (n-1)d. Used for: linear growth, regular intervals. Geometric sequences have a constant ratio (r) multiplied between terms: aₙ = aₙ₋₁ × r. Pattern: multiply by same amount each time. Example: 2, 6, 18, 54, 162 (multiply by 3). Exponential growth. Explicit: aₙ = a₁ × r^(n-1). Used for: exponential growth/decay, compound interest. Both are fundamental sequence types with distinct patterns and applications.

Question 10

Can you have a recursive formula with more than one previous term?

Accepted Answer

Yes! Recursive formulas can use multiple previous terms: Two terms: Fibonacci sequence aₙ = aₙ₋₁ + aₙ₋₂. Each term uses two previous terms. Example: 1, 1, 2, 3, 5, 8, 13... Three terms: Tribonacci aₙ = aₙ₋₁ + aₙ₋₂ + aₙ₋₃. Each term is sum of three previous. Example: 0, 0, 1, 1, 2, 4, 7... General: aₙ can depend on any number of previous terms. The more terms used, the more initial conditions needed. These are called recurrence relations of higher order. They're used in advanced mathematics, physics (differential equations), and computer science (dynamic programming). Each type creates unique patterns and has specific applications.

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