Induction (1 Viewer)

[velox]

Yes its fairly easy, but im having a mental block looking at the solution to this problem

Use mathematical induction to prove that for every positive integer n, (2^(n+2)) + (3^(3n) is divisible by 5.

 

[withoutaface]

Assume that

2^(k+2)+3^3k=5p(where p is an integer)

RTP
2^(k+3)+3^(3k+3)=2(5p-3^3k)+27*3^3k=10p+25*3^3k=5(2p+5*3^3k)

∴ divisible by 5

 

[velox]

how we get the 5p into the 2nd algebra manipulation? I dont understand the setting out.....fuck this is pissing me off, i had it yesterday, now it seems like i have never done it.

 

[gordo]

didn;t u do the hsc this year

 

[acmilan]

S(n): (2^(n+2)) + (3^(3n))

for n = 1, S(1) = 8 + 27 = 35 which is divisible by 5 hence S(1) is true

Assume true for n = k
s(k): (2^(k+2)) + (3^(3k)) = 5M (where M is any integer(

To prove n = k + 1 is also true

s(k+1):
(2^(k + 2 + 1)) + (3^(3k + 3))
= (2.2^(k + 2)) + (3^(3k).3^3)
= (2.(5M - 3^(3k)) + (27.3^(3k)) [since 2^(k+2) = 5M - 3^(3k) from s(k)]
= 10M - 2.3^3k + 27.3^3k
= 10M + 25.3^(3k)
= 5(2M + 5.3^(3k))
= 5N (where N = (2M + 5.3^(3k)) which is an integer since M and 3^3k are integers)

hence true for s(k+1)

the 5M comes in because its was substituted from s(k) into s(k+1)

 

[velox]

didn;t u do the hsc this year

Nope, why is everyone thinking that?

 

[Numero Uno]

Nope, why is everyone thinking that?

Mate u do 4 unit .. this should be easy for u

 

[velox]

Technically i dont, cos we dont start til next yr. But yeah it is easy. Read first post.

 

[acmilan]

Technically i dont, cos we dont start til next yr. But yeah it is easy. Read first post.

Next year? By the end of first term in year 12 my school had done the first two topics, tested us in a topic test and i dropped the course - all this in the first term

 

[velox]

well we finish 3u this term, and do 4u next yr.

 

[acmilan]

That explains it then, we finished 3 unit about end of term 1

 

[Numero Uno]

well we finish 3u this term, and do 4u next yr.

Huh ? wut school is this ?

 

[velox]

High

ten chars

 

[seremify007]

whats 'high'?

 

[Raginsheep]

Sydney I think...

 

[velox]

yep
ten chars