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.root-post .like-component { align-items: center; } .root-post .like-button { margin-left: 10px; } WOLFRAM COMMUNITY 1 | 286 Views | 2 Replies | 5 Total Likes View groups... Follow this post Share Share this post: GROUPS: Mathematics Algebra Equation Solving Wolfram Language

How to find the value of f(2023)+f(2024) ?

Lee Tao Lee Tao Lee Tao Posted 13 days ago .root-post .like-component { align-items: center; } .root-post .like-button { margin-left: 0; }

Given that the function f(x) has a domain of R and for any x,y∈R , the equation f(x−y)⋅f(x+y)=(f(x)−f(y))⋅(f(x)+f(y)) always holds true. Moreover, it is given that f(1)=2 and f(2)=0 . We are asked to find the value of f(2023)+f(2024) .

I tried this method and did not succeed in finding it.

RecurrenceTable[{f[x - y] f[x + y] == (f[x] - f[y]) (f[x] + f[y]), 
  f[1] == 2, f[2] == 0}, f, {x, 2024}, {y, 2024}]
POSTED BY: Lee Tao 2 Replies Sort By: .root-post .like-component { align-items: center; } .root-post .like-button { margin-left: 0; } 2 Jim Baldwin Jim Baldwin Jim Baldwin Posted 12 days ago

A brute force way is to plug in integer values for $x$ and $y$ and you'll find that $f(x)=0$ for even $x$. For odd $x$ you'll find that $f(x)=-2 i^{x+1}$. One such real valued function that has such properties is $\sin \left(\frac{\pi x}{2}\right)$.

f[x_] := Sin[\[Pi]  x/2]
f[x - y]  f[x + y] == (f[x] - f[y])  (f[x] + f[y]) // FullSimplify
(* True *)
POSTED BY: Jim Baldwin .root-post .like-component { align-items: center; } .root-post .like-button { margin-left: 0; } 2 Gianluca Gorni Gianluca Gorni Gianluca Gorni, University of Udine Posted 12 days ago

You could start by proving that f[n] must be zero for all even n.

POSTED BY: Gianluca Gorni
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