help me plz

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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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UMM OK LOL ?
wait does that make sense? lol
Cus like our teacher was quite strict on following the correct setup and all
YES IT DOES BUT I WAS LIKE CHANGING THE OAGE AND SAW ALL THAT I WAS LIKE WTH WHERE DID THAT COME FROM HAHA
PAGE
hahaahahhaa
ok so go on
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)
wait what was step one again haha sorry
hahah no problem, step 1 is to test the condition
so because we are asked to prove for all positive integers, its easiest to test for n =1
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
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
ok so it would look like umm shiz hah i for got ho to write it haha i jsut ad it sorry
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
omg your so awsome im fanning you haah
hahaha thanks lol I just like helping whenever I can :)
ok kool cuz i have like three or four more problems haha
oh I can try and help you with those later, but I gotta after this one, sorry :/
just having a bit of a headache, need rest haha
ok kool imm bee done for tonight after this one to maybe idk haha and thnx
hahah okay no problem, yeah its night for you guys, I live in the southern hemisphere so different time zones xD
i do to haha i live in south carolina
oh I live in australia so yeah far from US haha
well you said southren hemisphere soo i thought the south of the usa opps my bad hah
yeah I wasn't clear lol
but ik the times my bff is from newzealand
ohh coool!
yeah haha ok so is this problem done or nah haha
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)
do you see what I did there?
yeah i got it this time haha
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)
yeah
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)
does that make sense?
yes sir ma'am sir ma'am sir haha
hahaha okay, but we already know that 8 + 16 + 24 + ... + 8k = 4k(k + 1) (from when we let n = k)
yeah
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
yess
and our goal now it to make the LHS look like the RHS
omg this is so long haha
hahah ikr thats why they like a formally written proof
so we can make the LHS look like the RHS through factoring out the x + 1
this is why this is my last class and im done with school haha
hahahahaha well school isn't that bad
I used to hate it, but you miss it once you leave
well I'm told I should be missing it but meh
i will never haha
haha you never know
anyway so once we get the LHS to look like the RHS we can move onto step 3
ok i really need to go to my next problem so wnot to be rood but whats next mate ahah
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
but yea just make sure you write up the proof with all your steps in a formal way in a test
hat should i exactly google
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"
and thats your final step
thnx on two my next headache haha
haha no problem :) Just post them up and Ill try and take a look a bit later

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