brother dave explains it all
this is my brother, trying to explain my math homework to me. if you ever think about taking advanced pre-cal/algebra 3/trig....DONT!
kATy BirD ALoha: 8+11+14+...+(3n+5)=(3n^2+13n)/2
dave: what formulas have you learned?
kATy BirD ALoha: we havent
dave: like, have you learned what the sum of an arithmetic series is?
kATy BirD ALoha: yeah
kATy BirD ALoha: that is a mathematical induction problem
dave: ah
dave: ok
kATy BirD ALoha: we are supposed to prove it is true for K and K+1
dave: no, first you prove it for the base case, that is n=1 (since 3*1 + 5 = 8)
kATy BirD ALoha: yeha we are supposed to assume its rtue
dave: it's easy enough to check, just put one into the right side
kATy BirD ALoha: it works
dave: ok, so you assume sum(3n+5, 1, k) = (3k^2+13k) /2
kATy BirD ALoha: ok
dave: then you have to show the sum(3n+5,1,k+1) is equal to the right side for n=k+1
kATy BirD ALoha: do i put in k AND k+1?
kATy BirD ALoha: like 3k+5+3(k+1)+5
dave: well, the idea is this
dave: you assumed the sum from 1 to k is the (3k^2+13k)/2, right?
kATy BirD ALoha: ok
dave: and the sum from 1 to k+1 is the sum of 1 to k plus the k+t'th term
dave: k+1'th
kATy BirD ALoha: i dont get it
dave: alright, you take a sum by adding a series of terms, right?
kATy BirD ALoha: 1 to k=3k^2+13k/2
1 to k+1=3(k+1)^2+13(k+1)/2
kATy BirD ALoha: ?
dave: well, you have to show the second part
dave: think about what a sum is
dave: take for instance sum(n,1,2)
kATy BirD ALoha: we have to show it works for k and the term after that
kATy BirD ALoha: where n=k and n=any number
dave: if you use the assumption that k works, and then show that k+1 works with that assumption, you've shown that it's true for any number
dave: because you can take k as the base case (n=1)
dave: and then compute n=2
dave: by saying k=1, so k+1=2
kATy BirD ALoha: so....heres what my problem looks like so far:
kATy BirD ALoha: 8+11+14+...+(3n+5)=(3n^2+13n)/2
3k+5=(3k^2+13k)/2
3k+5+3(k+1)+5=(3(k+1)+13(k+1))/2
dave: no, (3k^2+13k)/2 doesn't equal 3k+5
kATy BirD ALoha: yes it does
dave: it equals (3(1)+5)+(3(2)+5)+(3(3)+5)+...+(3k+5)
kATy BirD ALoha: then what the hell is the right side?
dave: 3k^2+13k / 2
kATy BirD ALoha: what is it there for?
dave: it's equivalent to the sum you're given
dave: look at it this way f(x) = sum(3n+5,1,x) = (3x^2 + 13x)/2
kATy BirD ALoha: i do not understand this at all
dave: let's look at some small values of k
dave: say k=2
dave: so the sum(3n+5,1,k) = sum(3n+5,1,2) = 3(1)+5+3*2+5
dave: which = 8+11 = 19
dave: right?
kATy BirD ALoha: ok
dave: now, what's (3k^2+13k)/2 when k=2?
kATy BirD ALoha: 12+26=38
kATy BirD ALoha: /2
dave: /2
kATy BirD ALoha: 19
dave: chi-ching
kATy BirD ALoha: how do i show that with variables?
dave: induction
dave:
kATy BirD ALoha: thank you captain obvious
dave: alright, so what's sum(3n+5,1,k) where k=3?
dave: hint, use the sum from 1 to 2
kATy BirD ALoha: 22
dave: no, remember the last term is 3n+5
kATy BirD ALoha: 14
dave: that's the value of the last term
dave: you need to add it to the sums of the previous terms
kATy BirD ALoha: what was the first one?
kATy BirD ALoha: 14, 19, ?
dave: the sum of the first two terms we worked out was 19
kATy BirD ALoha: so 33
dave: right
kATy BirD ALoha: what does that mean?
dave: sum(f(n),1,k+1) = sum(f(n),1,k)+f(k+1)
kATy BirD ALoha: so?
dave: in this case, f(n) = 3n+5
dave: k=2
kATy BirD ALoha: k+1=3n+5
kATy BirD ALoha: k=3n+5
dave: no, f(n) = 3n+5
dave: f(k) = 3k+5
dave: f(k+1) = 3(k+1) + 5
kATy BirD ALoha: you are confusing me more
dave: forget about the f stuff then
kATy BirD ALoha: im going to fail this clas
dave: chill out, kate
dave: you're almost there
kATy BirD ALoha: i dont understand a word of what you just said
kATy BirD ALoha: and im already failing the class, tomorrows quiz will only make it worse
dave: don't worry about it
dave: look
dave: work these out on paper sum(x,1,3), sum(x,4,5)
kATy BirD ALoha: i dont know what that means!
kATy BirD ALoha: we didnt learn that
dave: the sigma stuff?
dave: if you didn't, don't worry about it
dave: going back to your problem, then
dave: 8+11+14+17+...+(3n+5)
kATy BirD ALoha: =
dave: don't worry about that yet
kATy BirD ALoha: ok
dave: what does it equal when n=6?
kATy BirD ALoha: 8+11+14+...+23
dave: and the ...?
kATy BirD ALoha: 8+11+14+17+20+23
kATy BirD ALoha: 93
dave: right
dave: now
dave: you see how each of those terms is the result of multiplying a number by three and then adding five?
kATy BirD ALoha: no, its just adding three
dave: the first term is 3*1+5, the second is 3*2+5
kATy BirD ALoha: ok
dave: does that make sense?
kATy BirD ALoha: yes
dave: the f stuff i was talking about earlier is a way of notating this
dave: when i say f(n) = 3n+5, i mean that if i put a number in for n, say 3, f(n) = the result of applying the function
kATy BirD ALoha: i dont care
dave: so f(3) = 3*(3)+5
kATy BirD ALoha: that will just confuse me
dave: ok then
dave: anyway, we figured out that when n=6, the sum is 8+11+14+17+20+23, or 93
dave: now, when i ask you to find the sum when n=7, what do you do?
kATy BirD ALoha: 93+3*7+5?
dave: yay
kATy BirD ALoha: 119:
dave: it's 8+11+14+..+23+3*7+5
dave: you already new the first part was 93, and then you added the next term
dave: is this making sense?
kATy BirD ALoha: yes
dave: that's like, fundamental to understand before we go on to applying it to induction
kATy BirD ALoha: i get it
dave: ok
dave: now
dave: on the right side, you're given some expression, and you're told it's equal to what's on the left (the 8+...)
kATy BirD ALoha: yeah
dave: and you need to use induction to show that's the case
dave: ok?
dave: hello?
kATy BirD ALoha: sorry
kATy BirD ALoha: needed agua
kATy BirD ALoha: right
dave: ok
dave: trying to regain my train of thought
dave: ok, for kicks, plug 6 into the expression on the right
kATy BirD ALoha: 93
dave: you see how what's on the right is different from what's on the left?
kATy BirD ALoha: no...it is the same thing
dave: yeah, but you didn't have to add a bunch of terms to get it
kATy BirD ALoha: yeah
kATy BirD ALoha: but how do you prove they are the same?
dave: guess
kATy BirD ALoha: we arent allowed to plug in #s
dave: no, induction
kATy BirD ALoha: i dont know wha tinduction is
kATy BirD ALoha: thats the problem
dave: alrighty
dave: induction is the three steps you've learned about: show it's true for a base case, assume it's true for some value k, and then using what you assumed in the first part, show that it's true for k+1
dave: so
dave: 8+11+14+...+3k+1 = (3k^2+13)/2
kATy BirD ALoha: 3k+1? what happened to 3k+5?
dave: typo
dave: what you need to show is that 8+11+14+...+3k+5+3(k+1)+5 = (3(k+1)^2 + 13(k+1))/2
kATy BirD ALoha: thats what i asked you in the first place!
dave: right, now about showing it
dave: the way i wrote the third part you should be able to figure it out
kATy BirD ALoha: make them the same on each side?
dave: yes
dave: what substitution can you make to get rid of all the terms on the left?
kATy BirD ALoha: 3k^2+13k/2
dave: so what does the left side now equal?
kATy BirD ALoha: 3k^2+13k/2+3(k+1)+5
dave:
dave: does it click now?
kATy BirD ALoha: sort of
dave: work through it, make sure you can show they're equal, then i'll give you another problem
kATy BirD ALoha: well i have moer
dave: i want to see you work it out
dave: more or less
dave: you finish that one?
kATy BirD ALoha: nooo!
kATy BirD ALoha: mom and dad just got home and they bought me a milkshake
dave: tell them to put it in the fridge until you've learned this shit!
kATy BirD ALoha: hey!
kATy BirD ALoha: ill drink my damn milkshake when i please
kATy BirD ALoha: i have:
kATy BirD ALoha: 3k^2+3k+21=3k^2+19k+16
kATy BirD ALoha: is that right so far?
dave: er, one sec
kATy BirD ALoha: can i just factor them the same?
dave: the coefficients aren't equal
dave: it's wrong
kATy BirD ALoha: what if they factor?
kATy BirD ALoha: i dont know what i did wrong
dave: i'm checking
kATy BirD ALoha: one sec...potty break
dave: right side is ok
dave: not sure what you did on the left
dave: wait
kATy BirD ALoha: multiplied both sides by 2
kATy BirD ALoha: oh wait
kATy BirD ALoha: i have to do both terms
kATy BirD ALoha: it works now
dave: what'd you get for the left side?
kATy BirD ALoha: 3k^2+19k+16
dave: right
dave: you shouldn't have to factor anything, just move around the terms until both sides are equal
AHHHHHHHHH! and thats only ONE problem. i have a sheet with FIFTEEN OF THEM on it!!!!!
