91Ó°ÊÓ

A recursive relationship is a connection between elements of a sequence or set where the next element is defined in terms of its predecessors. In this exercise, we analyze the provided function condition \[ \sum_{r=1}^{n} r f(r) = n(n+1) f(n) \]This introduces a recursive relationship for the values of the function \( f \). Recursive relationships are powerful as they can help us deduce unknown values based on previous known values.
In the example given, we used the known value \( f(1) = 1 \) to find \( f(2) \) through substitution and algebraic manipulation. By simplifying:

• \( f(1) + 2f(2) = 6f(2) \)
• \( 1 = 4f(2) \)

We found \( f(2) = \frac{1}{4} \). This process shows how recursive relationships enable us to build solutions step-by-step through logic based on initial conditions.

Proof by induction is a mathematical technique used to prove a statement is true for all natural numbers. The process typically involves two steps:

• Base Case: Verify the statement for the initial value, usually \(n=1\).
• Inductive Step: Assume the statement holds for some integer \(k\) and then prove it holds for \(k+1\).

In our scenario, starting from the base case \( f(1) = \frac{1}{1^2} = 1 \), we assumed that \( f(k) = \frac{1}{k^2} \) holds for some \(k\). Then, we needed to show it was also true for \( k+1 \):\[ \sum_{r=1}^{k+1} \frac{1}{r} = \frac{k+2}{k+1} \]Adding \( \frac{1}{k+1} \) to both sides of assumed equations allowed us to confirm that the relationship holds for \( k+1 \) as well.
Induction gives a structured approach to proving recursive hypotheses, reinforcing our understanding and prediction of patterns within mathematical solutions.

A functional equation is an equation that specifies a function in implicit form, which means it describes the function's behavior rather than providing an explicit formula. The functional equation provided in this exercise is:\[ \sum_{r=1}^{n} r f(r) = n(n+1) f(n) \]This requires finding a function \( f \) that satisfies this equation for all positive integers \( n \). Such functional equations allow us to explore deep properties of functions.By deriving insights from step-by-step substitution and simplification, we hypothesized and eventually proved using induction that \( f(n) = \frac{1}{n^2} \) satisfies the functional equation. The intuitive leap into trying \( \frac{1}{n^2} \) came from observing patterns in the solutions.Functional equations are crucial because they open pathways to understanding complex relationships in a simplified way, leveraging known properties and relationships to solve problems or uncover new functional forms.