Guess the number of triangles in the graph of as a function of n. ( has nested occurrences of .) When you know the number of triangles, you know all the characteristics of each triangle (first the altitude and the base length, from which you can find the side lengths since the triangles are isosceles).
Based on the graphs of for small values of n, make a hypothesis about the relationship between n and the number of triangles in the graph of . I am not sure how strictly you need to prove this hypothesis, but going from the graph of to that of should give you the idea of the proof of the hypothesis.
Hello, Mhmh96!
What a strange problem . . . bizarre!
I did a lot of sketching and muttering
. . and I think I've understood the composite functions.
If I'm wrong, my apologies for wasting your time . . .
has this graph.
It is an isosceles triangle with base 1 and height 1.Code:| 1+ - - - - - - - * | * * | * * | * * | * * | * * | * * | * * - - * - - - - - - - : - - - - - - - * - - 0 1/2 1
The slanted side has length . . . and there two of them.
Hence: .
has this graph.
It has two isosceles triangles with base 1/2 and height 1.Code:| 1+ - - - o - - - - - - - o | o o o o | o o o o | o o o o | o o o o | o o o o | o o o o |o o o o - - o - - - : - - - o - - - : - - - o - - 0 1/2 1
The slanted side has length: . . . and there are four of them.
Hence: .
has this graph.
It has four isosceles triangles with base 1/4 and height 1.Code:| 1+ - * - - - * - - - * - - - * | | * * * * * * * * | | * * * * * * * * | |* * * * * * * * | - - * - - - * - - - * - - - * - - - * - - 0 1/2 1
The slanted side has length . . . and there are eight of them.
Hence: .
I conjecture that: .
And therefore: .