Proving f(n) = 5 x 3^n - 4 with Mathematical Induction

[BasilBrush]

Homework Statement

Let f(n + 1) = 3f(n) + 8, with f(1) = 11. Prove by induction that f(n) = 5 x 3^n - 4.

Homework Equations

The Attempt at a Solution

I don't even know where to start! Any help would be appreciated. Thanks. :-)

 

[Mentallic]

Well since it gives you the base case, f(1)=1, you can show it's true for the base case n=0.

Now you assume it is true for n=k, that is,
3f(k)+8=5.3^k-4

Now prove it true for n=k+1,

3f(k+1)+8=5.3^{k+1}-4

 

[hunt_mat]

You don't really need induction, the recurrence relation is given by:
<br /> a_{n+1}=3a_{n}+8<br />
This suggests you look for a solution of the form:
<br /> a_{n}=\alpha\cdot 3^{n}+\beta<br />
For some \alpha ,\beta, then it is just a matter of plugging this into the equation to obtain values for \alpha and \beta.

Mat

 

[BasilBrush]

Well since it gives you the base case, f(1)=1, you can show it's true for the base case n=0.

Now you assume it is true for n=k, that is,
3f(k)+8=5.3^k-4

Now prove it true for n=k+1,

3f(k+1)+8=5.3^{k+1}-4

Nice one, thanks!