Problem 6 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(f(n+1)+f(n-1)=2 f(n)\) and \(f(0)=0\), then \(f(n), n \in N\) is (a) \(n f(1)\) (b) \((f(1))^{n}\) (c) 0 (d) None of these

Step-by-Step Solution

Answer

The function is \(f(n) = nf(1)\), represented by choice (a).

Step 1: Understand the Recurrence Relation

The exercise gives a recurrence relation: \(f(n+1) + f(n-1) = 2f(n)\). This means the function is defined in a way such that the next term and the previous term add up to twice the current term.

Step 2: Consider Initial Condition

We are given \(f(0) = 0\). This initial condition will help determine specific values of the function.

Step 3: Assume a Possible Solution Form

Consider if a sequence like polynomial can satisfy the relation. Assume \(f(n) = a_n\), where \(a_n\) could be in terms of an explicit formula for each term.

Step 4: Verify with Basic Terms

Calculate a few terms:- \(f(1) = f(1)\).- \(f(2) = 2f(1) - f(0) = 2f(1)\).It appears linear because each result seems to build linearly depending on previous terms. Assume a linear progression like \(f(n) = nf(1)\).

Step 5: Inductive Verification

Verify the assumed form \(f(n) = nf(1)\) by substituting it into the recurrence. Assume true for \(n\), check for \(n+1\): \\(f(n+1) = (n+1)f(1)\), which requires \((n+1)f(1) + (n-1)f(1) = 2nf(1)\) and simplifies to the same form. The relation holds by induction.

Step 6: Conclude the Sequence

The function follows \(f(n) = nf(1)\), satisfying the equation. Connect it to answer choices: (a) \(nf(1)\) exactly describes the sequence.

Key Concepts

Initial Conditions in Recurrence Relations

In mathematics, especially in the study of recurrence relations, initial conditions are crucially important. They act as the starting point or "anchor" for the sequence of values generated by the recurrence formula. These conditions determine the specific solution of a recursion among its general possible solutions.

For example, in the exercise, the given initial condition is \(f(0) = 0\). When combined with the recurrence relation \(f(n+1) + f(n-1) = 2f(n)\), this condition helps in defining the exact structure of the sequence function. Without such a starting point, the sequence could have infinitely many solutions, making it challenging to determine a unique function.

Initial conditions help in ensuring that the function generated aligns accurately with the context of the problem. As seen in this exercise, knowing that \(f(0) = 0\) helps in deducing the functional form \(f(n) = nf(1)\) by calculating the subsequent terms in the sequence.

Inductive Reasoning in Solving Recurrence Relations

Inductive reasoning is a logical process that involves making generalizations based on specific observations. In mathematics, particularly in dealing with recurrence relations, induction is a powerful tool used to establish proof of a sequence's behavior or formula.

Here is how it works:

• **Base Case:** Verify that the formula holds for the first term or initial condition, which in our case is \(f(0) = 0\).
• **Inductive Step:** Assume that the formula works for some arbitrary term \(n\) (known as the inductive hypothesis).
• Use this hypothesis to prove that the formula then holds for the \(n+1\) term.

In the exercise, once a plausible form of the solution \(f(n) = nf(1)\) was proposed, inductive reasoning was used: assuming \(f(n) = nf(1)\) is correct, one checks whether \(f(n+1) = (n+1)f(1)\) satisfies the recurrence, which confirms that the formula is consistent for all natural numbers \(n\). This shows the reliability of the function described and complies with the recurrence relation.

Understanding Sequence Functions

A sequence function is a rule or formula that determines the terms of a sequence in relation to their position within the sequence.

In this exercise, the sequence function is dictated by the recurrence relation \(f(n+1) + f(n-1) = 2f(n)\) and the initial condition \(f(0) = 0\). To uncover the underlying sequence, it helps to examine the nature of recurrence and calculate the first few terms to perceive any noticeable pattern.

This process involves:

• Assuming a possible form like \(f(n) = nf(1)\), which often involves imagining typical sequences (such as linear, quadratic, etc.) that could fit the given relation.
• Substituting these guessed forms back into the original recurrence relation to verify their validity.

In this case, upon substituting \(f(n) = nf(1)\), the relation holds true, meaning the linear assumption was correct for the sequence. This culminates in identifying \(f(n) = nf(1)\) as the solution to the recursive relation, thus defining the function of the sequence fully.