Asked by Tesla
Which sequence can be generated from the formula f(x + 1) = One-half(f(x))?
All Answers 1
Answered by
GPT-5 mini
AI
All sequences where each term is half the previous term — i.e. geometric sequences with common ratio 1/2.
If x is an integer and you know f(0) (or f(1)), the explicit formula is
- f(n) = f(0)*(1/2)^n (or equivalently f(n) = f(1)*(1/2)^{n-1}).
Examples:
- If f(0)=16: 16, 8, 4, 2, 1, 1/2, ...
- If f(1)=1: 1, 1/2, 1/4, 1/8, ...
(If x is allowed to be real, the general solution is f(x) = (1/2)^x · p(x) where p is any function with period 1; for continuous p this reduces to f(x)=C·(1/2)^x.)
If x is an integer and you know f(0) (or f(1)), the explicit formula is
- f(n) = f(0)*(1/2)^n (or equivalently f(n) = f(1)*(1/2)^{n-1}).
Examples:
- If f(0)=16: 16, 8, 4, 2, 1, 1/2, ...
- If f(1)=1: 1, 1/2, 1/4, 1/8, ...
(If x is allowed to be real, the general solution is f(x) = (1/2)^x · p(x) where p is any function with period 1; for continuous p this reduces to f(x)=C·(1/2)^x.)
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