Question

help me plz

Answer 1

@surjithayer

Answer 2

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?

Answer 3

UMM OK LOL ?

Answer 4

wait does that make sense? lol

Answer 5

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

Answer 6

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

Answer 7

PAGE

Answer 8

hahaahahhaa

Answer 9

ok so go on

Answer 10

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)

Answer 11

wait what was step one again haha sorry

Answer 12

hahah no problem, step 1 is to test the condition

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

omg your so awsome im fanning you haah

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

just having a bit of a headache, need rest haha

Answer 23

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

Answer 24

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

Answer 25

i do to haha i live in south carolina

Answer 26

oh I live in australia so yeah far from US haha

Answer 27

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

Answer 28

yeah I wasn't clear lol

Answer 29

but ik the times my bff is from newzealand

Answer 30

ohh coool!

Answer 31

yeah haha ok so is this problem done or nah haha

Answer 32

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)

Answer 33

do you see what I did there?

Answer 34

yeah i got it this time haha

Answer 35

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)

Answer 36

yeah

Answer 37

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)

Answer 38

does that make sense?

Answer 39

yes sir ma'am sir ma'am sir haha

Answer 40

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

Answer 41

yeah

Answer 42

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

Answer 43

yess

Answer 44

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

Answer 45

omg this is so long haha

Answer 46

hahah ikr thats why they like a formally written proof

Answer 47

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

Answer 48

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

Answer 49

hahahahaha well school isn't that bad

Answer 50

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

Answer 51

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

Answer 52

i will never haha

Answer 53

haha you never know

Answer 54

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

Answer 55

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

Answer 56

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

Answer 57

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

Answer 58

hat should i exactly google

Answer 59

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"

Answer 60

and thats your final step

Answer 61

thnx on two my next headache haha

Answer 62

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