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:
$$a_{n+1}=3a_{n}+8$$
This suggests you look for a solution of the form:
$$a_{n}=\alpha\cdot 3^{n}+\beta$$
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!