Question 1196880: If f(x+1)-7=f(x), find f(x)
Answer by ikleyn(52752)   (Show Source):
You can put this solution on YOUR website! .
If f(x+1)-7 = f(x), find f(x).
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If f(x+1)-7 = f(x), it implies that
f(x+1) = f(x) + 7. (1)
If consider "x" as a variable in integer numbers (as an "index"), then equation (1)
defines an arithmetic progression with the common difference 7.
Had a value of f(x) be given in the problem for some integer value of x,
then equation (1) would determine the progression in a UNIQUE way.
But since the value of f(x) is not given in the problem for some x,
we can say nothing more about this arithmetic progression.
There are infinitely many arithmetic progressions described by this formula.
If consider "x" as a variable in real numbers (a continuous variable), then equation (1)
defines a straight line with the slope of 7:  = 7.
Had a value of f(x) be given in the problem for some real value of x,
then it would determine the straight line in a UNIQUE way.
But since f(x) is not given in the problem for some x, there are infinitely many
such straight lines described by this equation (with the slope 7).
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Solved, answered and explained.
That is all what Math can tell you about f(x) defined by the given equation.
With the given information in the post, it is IMPOSSIBLE " to find f(x) ".
What is only possible is to determine a class (or a type) of a sequence f(x)
(as an arithmetic progression in discrete case) or a function f(x) (as a linear function).
So, if to understand each word literally in the post, this post is not SELF-CONSISTENT:
it requires to do something, that can not be done.
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