Induction Question (1 Viewer)

[ShAzABoB]

Can someone please show me how to do this? i can never get to the end of induction questions >.<

Prove 2<SUP>n+2 </SUP>+ 3<SUP>3n</SUP> is divisible by 5

 

[vafa]

Solution:

View attachment 13868

 

[vafa]

I forgot to write first step, you shoulf first address step 1.

 

[SoulSearcher]

Assuming that it is for all n > 1,
n = 1,
2³ + 3³ = 8 + 27 = 35, which is divisible by 5
Therefore assume that k is an interger for which the statement is true, that is
2^k+2 + 3^3k = 5m, for some integer m,
i.e. 3^3k = 5m - 2^k+2
Now we have to prove the statement for n = k+1, that is
2^k+3 + 3^3k+3
= 8*2^k + 27*3^3k
using the induction hypothesis,
= 8*2^k + 27(5m - 2^k+2
= 8*2^k + 27*5m - 108*2^k
= 27*5m - 100*2k²
= 5(27m - 20*2^k), which is divisible by 5, therefore the statement is true for all n > 1 by mathematical induction.

 

[ShAzABoB]

ummmm how did you go from:
2.2<SUP>k+2 </SUP>+ 3<SUP>3k</SUP>.3<SUP>3</SUP><?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>
<o> </o>
to <o></o>
<o> </o>
2(2<SUP>k+2 </SUP>+ 3<SUP>3k</SUP>) + 25.3<SUP>3k</SUP>
<SUP></SUP>
sorry... im slow... but thank you!

 

[vafa]

let 3^3k=a and 2^k+2=b

2b+27a=(2b+2a)+25a=2(b+a)+25a

 

[ShAzABoB]

thank you!!! thats a bit simpler

 

[vafa]

I advise you to learn soulsearcher's solution because that is much more simpler.

 

[gamecw]

<SUP></SUP>
sorry... im slow... but thank you!

ur not slow, ur cute

 

[SoulSearcher]

ur not slow, ur cute

Haha, oh man ...

 

[ShAzABoB]

 

[~shinigami~]

You have "female" next to the Gender thing under your avatar so you better get used to it.

 

[gamecw]

You have "female" next to the Gender thing under your avatar so you better get used to it.

lol yes...

 

[Riviet]

gamecw must have seen her profile picture.

 

[~shinigami~]

I believe ShAzABoB is the one on the left.

 

[pLuvia]

Please stay on topic guys

 

[vafa]

Please stay on topic guys

Proof By Induction:

S(n): n people in this thread would like to stay on topic. when n is a natural number.

S(1): 1 person in this thread would like to stay on topic and this is true because Pluvia indicated that he would like to stay on topic and he/she is 1 person. Hence the statement is true for n=1

S(k): assumme k people in this thread would like to stay on topic.

s(k+1): k+1 people in this thread would like to stay on topic.

proof of s(k+1): k+1 consists of k people plus one person, from our assumtion k people would like to stay in this topic and from our s(1) one person would like to stay on topic (that is Pluvia) so K+1 people would like to stay on topic

Hece s(n) is true for n=1, it is then true for n=2 and hence for n=3 and so on for all natural numbers.

Conclusion:
Since the statement "n people in this thread would like to stay on topic" is true for 1 person and that is pLuvia, Then statement is true for n=2, it means 2 people would like to stay on topic and hence for n=3, it means 3 people would like to stay on topic and so on for all natural numbers, it means everyone in this thread would like to stay on topic.

 

[acmilan]

^^flawed proof

 

[gamecw]

gamecw must have seen her profile picture.

yea ive seen it so? :mad1:

I believe ShAzABoB is the one on the left.

Na i believe thats me

 

[ShAzABoB]

I believe ShAzABoB is the one on the left.

the one one the left??? thats not nice...

Secondly, pLuvia... my apologies... and my apologies to the other n people after him...

the rest of you... stay on topic please.... "Induction Question"

Lastly im not a boy! :mad1: