prove by induction

[chipatel87]

Suppose f0 = 5, f1 = 16, and fn = 7fn-1 - 10fn-2 for all n ≥ 2. Prove, using mathematical induction,that fn = 3 (2n) + 2 (5n) for all integers n ≥ 0.
the first step is to prove the logic f0 = 5
7(2-1)- 10(2-2) = 7

i have uploaded the image to show the problem better. I am stuck here and bad at the induction part. can some one help me?
there are more problems in the image also. If you are good at logic math. Help me with them please. thank you.

 

[galactus]

Show the first step. That f(0) is true.

Then, assume \(\displaystyle f_{k+1}=7f_{k}-10f_{k-1}\) is true.

Show that \(\displaystyle f_{k+1}=3\cdot 2^{k+1}+2\cdot 5^{k+1}\) is true.

\(\displaystyle f_{k+1}=7\left(3\cdot 2^{k}+2\cdot 5^{k}\right)-10\left(3\cdot 2^{k-1}+2\cdot 5^{k-1}\right)\)

\(\displaystyle =21\cdot 2^{k}+14\cdot 5^{k}-30\cdot 2^{k-1}-20\cdot 5^{k-1}\)

\(\displaystyle =21\cdot 2^{k}+14\cdot 5^{k}-15\cdot 2^{k}-4\cdot 5^{k}\)

\(\displaystyle =6\cdot 2^{k}+10\cdot 5^{k}\)

\(\displaystyle =3\cdot 2^{k+1}+2\cdot 5^{k+1}\)

 

[chipatel87]

union and intersection

Thank you sir. this cleryfies lot...tnx