Maths Induction question (1 Viewer)

[Dimsimmer]

3^(3n)+2^(n+2) is divisible by 5. I just only need the proof for n=k+1 part thanks.

 

[Templar]

For n=k+1

3^(3k+3)+2^(k+3)
=27*3^(3k)+2*2^(k+2)
=2(3^(3k)+2^(k+2))+25*3^(3k)

5|3^(3k)+2^(k+2); 5|25

Therefore divisible by 5.

 

[Riviet]

Hi there.
Assume 3^3k+2^k+2=5A, ie 3^3k=5A-2^k+2, where A is an integer
Prove for n=k+1, ie prove 3^k+3+2^k+3=5B, where B is an integer
LHS=3^k+3+2^k+3
=3²(5A-2^k+2)+2^k+3 by the assumption
=5x27A-108x2^k+8x2^k
=5x27A-100x2^k
=5(27A-20x2^k)
=5B, where B=27A-20x2^k
.: true for n=k+1
There you go.