Induction FUN (1 Viewer)

[243_robbo]

can anyone do question four - see attatchment (i scanned the whole thing so that i can refer to other questions if need be, and if anyone feels like some maths)

 

[pLuvia]

Hey what topics have you completed at school robbo? it seems your school is pretty fast

 

[243_robbo]

complex numbers, polynomials, integration, graphs, harder 3 unit circle geometry and induciton,


mainley cos for term 4 last year after hsc we had about 5 ext 2 classes a week owing tothe fact the whole class is the accelerants who did their hsc last year, and wd already covered 2 and 3 unit topics, but now we get only 2-3 ext 2 lessons a week.

 

[SeDaTeD]

"The result follows by induction" would suffice.

 

[pLuvia]

complex numbers, polynomials, integration, graphs, harder 3 unit circle geometry and induciton,


mainley cos for term 4 last year after hsc we had about 5 ext 2 classes a week owing tothe fact the whole class is the accelerants who did their hsc last year, and wd already covered 2 and 3 unit topics, but now we get only 2-3 ext 2 lessons a week.

Wow, covered a lot of topics already

 

[243_robbo]

This thread is a misnomer. Induction isn't fun.

You need to use strong induction for this question.

Show that the statement is true for n=1 and n=2

Assume that it is true for all 3<= n <=k

Then u_k-1 = 2 + 3^k-1
and u_k = 2 + 3^k

Using the recursion given,
u_k+1 = 4u_k-1 - 3u_k
=4(2 + 3^k) - 3(2 + 3^k-1)
=2 + 12*3^k-1 - 3*3^k-1
=2 +9*3^k-1
=2+ 3^k+1

as required.

Repeat the induction mantra, and you're done.

strong induction, eh?

ill give it a go, is that part of 4 unit syllabus by the way?

 

[Riviet]

I guess this is what one of the main distinctions between 4u and 3u induction is: 3u induction will only require you to make 1 assumption, whereas in 4u, you can be required to use more than one assumption to prove the statement true for n=k+1. Also you sometimes need to manipulate your assumption so you are able to use it in your proof.

 

[c0okies]

BUMP: could someone do question 11 from the picture? =]; thanx ahead

 

[_ShiFTy_]

I assume you would be able to do the 1st part cos thats just manipulation and stuff

Not sure about my answer for the 2nd part..

Assume
u_n = 9ⁿ⁺¹ -8n - 9 = 64k
Prove
u_n+1 = 9ⁿ⁺² -8(n+1) - 9 = 64M

u_n+1 = 9u_n + 64n + 64 = 64M

u_n+1 = 9(64K) + 64n + 64 = 64M

u_n+1 = 64(9K + n + 1) = 64M

Therefore divisible by 64?


Seems too simple to be correct tho



 

[c0okies]

LOL i was wondering abt the manipulation part; but now i get it rofl, no wonder nobody asked about it in past years

yea its correct, jus re-did it =)

thanx