# We have #f(n)# an string;#ninNN# such that #f(n+1)-f(n)=3f(n)# and #f(0)=-1/2#.How to express #f(n)# according to #n#?

Making

This is a homogeneous linear difference equation with generic solution

Substituting we have

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To express ( f(n) ) in terms of ( n ), we can solve the recurrence relation ( f(n + 1) - f(n) = 3f(n) ) with the initial condition ( f(0) = -\frac{1}{2} ).

First, let's rewrite the recurrence relation to isolate ( f(n + 1) ): [ f(n + 1) = 3f(n) + f(n) ]

Now, let's find a pattern:

[ \begin{align*} f(1) &= 3f(0) + f(0) \ &= 3\left(-\frac{1}{2}\right) - \frac{1}{2} \ &= -\frac{3}{2} - \frac{1}{2} \ &= -2 \end{align*} ]

[ \begin{align*} f(2) &= 3f(1) + f(1) \ &= 3(-2) - 2 \ &= -6 - 2 \ &= -8 \end{align*} ]

[ \begin{align*} f(3) &= 3f(2) + f(2) \ &= 3(-8) - 8 \ &= -24 - 8 \ &= -32 \end{align*} ]

From these computations, we observe that ( f(n) = -2 \times 3^n ).

So, the expression for ( f(n) ) according to ( n ) is ( f(n) = -2 \times 3^n ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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