help me plz

- anonymous

help me plz

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- anonymous

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- anonymous

@surjithayer

- FireKat97

For mathematical induction proofs, it is very important to follow through each step carefully, and our first step should be to test the condition. so we are asked to prove for all positive integers, so can you write out the test for a positive integer for step 1?

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- anonymous

UMM OK LOL ?

- FireKat97

wait does that make sense? lol

- FireKat97

Cus like our teacher was quite strict on following the correct setup and all

- anonymous

YES IT DOES BUT I WAS LIKE CHANGING THE OAGE AND SAW ALL THAT I WAS LIKE WTH WHERE DID THAT COME FROM HAHA

- anonymous

PAGE

- FireKat97

hahaahahhaa

- anonymous

ok so go on

- FireKat97

okay so after you test for your first step, you want to let n = k for your second step (another useless convention I suppose but it was necessary for us)
so its like...
Step 2, for n = k
8 + 16 + 24 + ... + 8k = 4k(k + 1)

- anonymous

wait what was step one again haha sorry

- FireKat97

hahah no problem, step 1 is to test the condition

- FireKat97

so because we are asked to prove for all positive integers, its easiest to test for n =1

- anonymous

ok how do i do that agin hah im sorry i was at base untill 8pm cuz my comander is an retriceso im very sleepy and slow haha

- FireKat97

haha no problem, so you can test this by saying;
test for n = 1
and then sub n = 1 into both the left and right hand sides and show that they are equal

- anonymous

ok so it would look like umm shiz hah i for got ho to write it haha i jsut ad it sorry

- FireKat97

hahah don't worry, so you just sub n = 1 into left and right hand side so
LHS = 8(1)
= 8
RHS = 4(1)(1 + 1)
= 4(2)
= 8
therefore LHS = RHS and the test is true for n = 1

- anonymous

omg your so awsome im fanning you haah

- FireKat97

hahaha thanks lol I just like helping whenever I can :)

- anonymous

ok kool cuz i have like three or four more problems haha

- FireKat97

oh I can try and help you with those later, but I gotta after this one, sorry :/

- FireKat97

just having a bit of a headache, need rest haha

- anonymous

ok kool imm bee done for tonight after this one to maybe idk haha and thnx

- FireKat97

hahah okay no problem, yeah its night for you guys, I live in the southern hemisphere so different time zones xD

- anonymous

i do to haha i live in south carolina

- FireKat97

oh I live in australia so yeah far from US haha

- anonymous

well you said southren hemisphere soo i thought the south of the usa opps my bad hah

- FireKat97

yeah I wasn't clear lol

- anonymous

but ik the times my bff is from newzealand

- FireKat97

ohh coool!

- anonymous

yeah haha ok so is this problem done or nah haha

- FireKat97

okay so lets continue with the question so
for step 2 we start off by letting n = k to get
8 + 16 + 24 + ... + 8k = 4k(k + 1)
then we want to let n = k + 1 and do a similar thing, subbing in K + 1 for all 'n's
so we get
Let n = k + 1
8 + 16 + 24 + ... + 8(k + 1) = 4(k + 1)(k + 1 + 1)

- FireKat97

do you see what I did there?

- anonymous

yeah i got it this time haha

- FireKat97

haha okay cool so now do you notice how we can rewrite the above line as
8(1) + 8(2) + 8(3) + ... +8(k + 1) = 4(k + 1)(k + 2)

- anonymous

yeah

- FireKat97

so following that trend, we can write 8(k) before the 8(k + 1)
so we get
8 + 16 + 24 + ... + 8k + 8(k+1) = 4(k + 1)(k + 2)

- FireKat97

does that make sense?

- anonymous

yes sir ma'am sir ma'am sir haha

- FireKat97

hahaha okay, but we already know that 8 + 16 + 24 + ... + 8k = 4k(k + 1) (from when we let n = k)

- anonymous

yeah

- FireKat97

so we can substitute that in, so rather than having
8 + 16 + 24 + ... + 8k + 8(k + 1) = 4(k + 1)(k + 2)
we can now have
4k(k + 1) + 8(k + 1) = 4(k + 1)(k + 2) instead

- anonymous

yess

- FireKat97

and our goal now it to make the LHS look like the RHS

- anonymous

omg this is so long haha

- FireKat97

hahah ikr thats why they like a formally written proof

- FireKat97

so we can make the LHS look like the RHS through factoring out the x + 1

- anonymous

this is why this is my last class and im done with school haha

- FireKat97

hahahahaha well school isn't that bad

- FireKat97

I used to hate it, but you miss it once you leave

- FireKat97

well I'm told I should be missing it but meh

- anonymous

i will never haha

- FireKat97

haha you never know

- FireKat97

anyway so once we get the LHS to look like the RHS we can move onto step 3

- anonymous

ok i really need to go to my next problem so wnot to be rood but whats next mate ahah

- FireKat97

and for step 3, we just need to kind of sum stuff up and say that by the law of mathematical induction.. blah blah blah (Im pretty sure you can google one of these statements, i don't remember them exactly) we have proven that the above statement is true for all positive integers

- FireKat97

but yea just make sure you write up the proof with all your steps in a formal way in a test

- anonymous

hat should i exactly google

- FireKat97

its actually just something along the lines of "According to the Principle of Mathematical Induction P(n) is true for any positive integer, as proven above"

- FireKat97

and thats your final step

- anonymous

thnx on two my next headache haha

- FireKat97

haha no problem :) Just post them up and Ill try and take a look a bit later

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