Problem 10 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If a function \(F\) is such that \(F(0)=2, F(1)=3\), \(F(n+2)=2 F(n)-F(n+1)\) for \(n \geq 0\), then \(F(5)\) is equal to (a) \(-7\) (b) \(-3\) (c) 7 (d) 13

Step-by-Step Solution

Answer

The value of \(F(5)\) is 13.

Step 1: Understand the problem

We are given a recursive function definition \(F(n+2)=2F(n)-F(n+1)\) with initial values \(F(0)=2\) and \(F(1)=3\). We need to find the value of \(F(5)\).

Step 2: Apply the recursive function for F(2)

Using the given formula, calculate \(F(2)\): \[ F(2) = 2F(0) - F(1) = 2(2) - 3 = 4 - 3 = 1 \]

Step 3: Calculate F(3)

Now, use \(F(1)\) and \(F(2)\) to find \(F(3)\):\[ F(3) = 2F(1) - F(2) = 2(3) - 1 = 6 - 1 = 5 \]

Step 4: Calculate F(4)

Use \(F(2)\) and \(F(3)\) to calculate \(F(4)\):\[ F(4) = 2F(2) - F(3) = 2(1) - 5 = 2 - 5 = -3 \]

Step 5: Calculate F(5)

Finally, use \(F(3)\) and \(F(4)\) to find \(F(5)\):\[ F(5) = 2F(3) - F(4) = 2(5) - (-3) = 10 + 3 = 13 \]

Key Concepts

Recurrence Relations

Recurrence relations are mathematical formulas used to define sequences of values using previous terms in the sequence. These relations are especially helpful in working with recursive functions, as they provide a way to express complex sequences without directly computing each element from scratch.
In the exercise, the recurrence relation is given by: \[F(n+2) = 2F(n) - F(n+1)\] This means that to find any term in the sequence beyond the first two known values, you rely on a fixed operation with the preceding two terms. It's like the blueprint of the sequence, detailing how each term is constructed from its predecessors.
When working with recurrence relations, always start by defining base cases, like \(F(0)\) and \(F(1)\), which serve as the foundation. From there, apply the recurrence formula to progressively build up to the desired term, in this case \(F(5)\). This method breaks down a potentially complicated problem into smaller, manageable steps.

Sequence Calculation

Sequence calculation is the practical application of recurrence relations to determine specific terms within a sequence. In this context, you compute each term step by step using the recurrence formula.
For the given function \(F\), we start by identifying the base values provided:

• \(F(0) = 2\)
• \(F(1) = 3\)

Using these, we apply the recurrence relation:

• \(F(2) = 2F(0) - F(1) = 2(2) - 3 = 1\)
• \(F(3) = 2F(1) - F(2) = 6 - 1 = 5\)
• \(F(4) = 2F(2) - F(3) = 2 - 5 = -3\)
• Finally, \(F(5) = 2F(3) - F(4) = 10 + 3 = 13\)

Each step relies upon the previous values, demonstrating how an entire sequence can be calculated if the recurrence is followed methodically.

Function Evaluation

Function evaluation in recursive functions involves using initial conditions and recurrence relations to determine the value of the function at a specific integer input. To evaluate the function means to recursively apply the given formula until you reach the desired term.
For functions like \(F(n+2) = 2F(n) - F(n+1)\), evaluating requires a systematic approach:- First, confirm the initial values are correct.- Use the recurrence relation to handle step-by-step calculations.- Keep track of each result, as they often serve as inputs for the next calculations.
This method ensures that you correctly derive \(F(5)\) as 13 by applying the recurrent formula iteratively from known values. Always double-check calculations at each step to prevent errors from propagating through the sequence.