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Math problem in senior high school: If f(x)=2x+ 1, what is f (……………………………………))? (there are n f's)
Solution:

f(f(x))= 2(2x+ 1)+ 1 = 4x+3

f[f(f(x))]= 2(4x+3)+ 1 = 8x+7

f { f[f(f(x))]} = 2(8x+7)+ 1 = 16x+ 15

Therefore:

f(f(f(……f(x)……))=

(2^n)x+(2^n)- 1

Let's prove it by mathematical induction.

(1) When n= 1

F(x)=2x+ 1, which is consistent with the known results;

When n=