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
By signing up, you agree to our Terms of Service and Privacy Policy
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 ).
By signing up, you agree to our Terms of Service and Privacy Policy
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.
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.
- How do you use the Nth term test on the infinite series #sum_(n=1)^oorootn(2)# ?
- How do you Use an infinite geometric series to express a repeating decimal as a fraction?
- Why does the integral test not apply to #Sigma (2+sinn)/n# from #[1,oo)#?
- Show that #lim_( x->a) (x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))=9/40 a^(-31/24)#?
- How do you find the positive values of p for which #Sigma n(1+n^2)^p# from #[2,oo)# converges?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7