Loser66
  • Loser66
Let \(\{a_n\}_{n=1}^\infty \) be defined recursively by \(a_1=1\) and \(a_{n+1}=\dfrac{n+2}{n}a_n\) for \(n\geq 1\)/ Then \(a_{30}\) is equal to?? A) (15)(30) B) (30)(31) C) \(\dfrac{31}{29}\) D) \(\dfrac{32}{30}\) E) \(\dfrac{32!}{30!2!}\) Please, help
Mathematics
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SOLVED
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
Loser66
  • Loser66
@Kainui @Michele_Laino
freckles
  • freckles
\[a_1=1 \\ a_2=\frac{3}{1}a_1=3(1)=3 \\ a_3=\frac{4}{2}a_2=2a_2=2(3)=6 \\ a_4=\frac{5}{3}a_3=\frac{5}{3}(6)=5(2)=10 \\ 1,3,6,10,... \text{ subtracting term from its previous term gives: } \\ 2,3,4,.. \\ \text{ doing that again we have } 1,1,1,... \\ \text{ so we have something \in the form } a_n=An^2+Bn+C \\ \text{ we could defined } a_0=0 \text{ since \it fits our pattern thingy } \\ \text{ so we could say } C=0 \\ a_n=An^2+Bn\] you can use the other terms in the sequence to find A and B this is the way I would think to do it there may be a quicker way (don't know about that though)
ganeshie8
  • ganeshie8
same thing but in reverse \[a_{n+1}~=~\dfrac{n+2}{n}\frac{n+1}{n-1}\cdots+\frac{1+2}{1} = \frac{(n+2)!}{n!2!}a_1~ =~ \binom{n+2}{2}a_1\]

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freckles
  • freckles
\the thingy you had before was correct since a1 is 1
freckles
  • freckles
but this is correct to :p
freckles
  • freckles
I didn't think to do all of that that is pretty neat
freckles
  • freckles
oh oh if you notice that the first term is 1 the second terms is 1+2 the third term is 1+2+3 the fourth term is 1+2+3+4 ... then you know the nth term is given by n(n+1)/2
freckles
  • freckles
so you can skip finding A and B the icky way
ganeshie8
  • ganeshie8
OMG! how triangular numbers popped up all off sudden!
freckles
  • freckles
I found the first few terms above
ganeshie8
  • ganeshie8
Ahh i wasn't thinking.. \(\large \binom{n}{2}\) is a triangular number...
freckles
  • freckles
Show \[\left(\begin{matrix}n+2 \\ 2\end{matrix}\right) \text{ is a triangular number for } n \ge 0\] \[\frac{(n+2)!}{2!(n+2-2)!}=\frac{(n+2)!}{2!n!}=\frac{(n+2)(n+1)n!}{2n!}=\frac{(n+2)(n+1)}{2}\]
Loser66
  • Loser66
I am sorry. My computer is crazy!!
Loser66
  • Loser66
I have a formula to find it, but I don't have my note with me right now. I will post it when I get home
ganeshie8
  • ganeshie8
just trying to see if the sequence in question can be converted easily to the familiar triangular numbers sequence form \(\large a_{n} = a_{n-1}+n\)
xapproachesinfinity
  • xapproachesinfinity
i got where ganesh got that formula \[a_{30}=\frac{31!}{2!29!}\] seems to me a bit of?
xapproachesinfinity
  • xapproachesinfinity
off*
xapproachesinfinity
  • xapproachesinfinity
Am i doing some kind of a mistake?
freckles
  • freckles
just need simplifying \[a_{30}=\frac{31 \cdot 30 \cdot 29!}{2 \cdot 29!}\]
xapproachesinfinity
  • xapproachesinfinity
yeah i did but does not look like it is giving the right answer
freckles
  • freckles
31*15
xapproachesinfinity
  • xapproachesinfinity
not on the choices
freckles
  • freckles
oh see what you are saying
xapproachesinfinity
  • xapproachesinfinity
i got the same thing
xapproachesinfinity
  • xapproachesinfinity
everything seems to me perfectly good given the pattern
freckles
  • freckles
oh loser made typeo
freckles
  • freckles
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0CCUQFjAB&url=https%3A%2F%2Fwww.ets.org%2Fs%2Fgre%2Fpdf%2Fpractice_book_math.pdf&ei=a-R1Vbv1N8fusAWXq4LQCg&usg=AFQjCNHO5Z7_9WxufkCzxQj2ggUgjYjnxg&sig2=NHd5ZQO0tokDsYeYHFKMAA&bvm=bv.95039771,d.b2w question 25
freckles
  • freckles
choice A is suppose to read (15)(31)
xapproachesinfinity
  • xapproachesinfinity
oh ok
freckles
  • freckles
yeah she is studying for the gre which she told me she was going to ace
xapproachesinfinity
  • xapproachesinfinity
i'm thinking or your question showing that (n+1)(n+2)/2 is triangular for n>0
freckles
  • freckles
(n+1)(n+2)/2 is triangular I thought
freckles
  • freckles
consecutive integers divided by 2 that is triangular right?
xapproachesinfinity
  • xapproachesinfinity
(n+1)(n+2)/2=(n+1)+n+......+1
xapproachesinfinity
  • xapproachesinfinity
yeah seems to be it is
freckles
  • freckles
ok you want to prove: \[\sum_{i=1}^{n}i=\frac{n(n+1)}{2} \\ \sum_{i=1}^{n+1}i=\frac{(n+1)(n+1+1)}{2}=\frac{(n+1)(n+2)}{2}\]
freckles
  • freckles
we can prove that by induction there is also one other way and that is to actually derive that equality sometimes I forget how to do that let me see I can derive it... *freckle's processor processing*
freckles
  • freckles
\[S(n)=\sum_{i=1}^{n}i \\ S(n+1)=\sum_{i=1}^{n+1}i \\ S(n+1)-S(n)=(n+1)\] I think it starts something like this
freckles
  • freckles
hmmm...
freckles
  • freckles
could look it up but don't want to cheat yet
xapproachesinfinity
  • xapproachesinfinity
hmm yeah i remember this
freckles
  • freckles
lol I don't remember
freckles
  • freckles
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/triNbProof.html
xapproachesinfinity
  • xapproachesinfinity
the algebra proof is the gauss's way :)
freckles
  • freckles
the proving the equality thing was easy by induction i just have a hard time remembering how to derive formulas like that
freckles
  • freckles
it does involve a bag of tricks
xapproachesinfinity
  • xapproachesinfinity
wow that GRE is not that easy haha
xapproachesinfinity
  • xapproachesinfinity
has a lot of questions that require some deep thinking lol
freckles
  • freckles
yep I don't remember it being that hard
xapproachesinfinity
  • xapproachesinfinity
you took one before?
freckles
  • freckles
there are questions on here that I'm not sure I can answer
freckles
  • freckles
yea about a decade ago
xapproachesinfinity
  • xapproachesinfinity
i see! I have never taking any standardized test yet
freckles
  • freckles
wow lucky
xapproachesinfinity
  • xapproachesinfinity
most of what is in that GRE is hard to me lol
xapproachesinfinity
  • xapproachesinfinity
what is the duration for such test? clearly this is not the exam it self?
xapproachesinfinity
  • xapproachesinfinity
it says 170 mins nearly 2 hour! for all that! that is a really frustrating hehe
freckles
  • freckles
Yeah. Very frustrating I bet.
anonymous
  • anonymous
Just offering another approach we can take. First set \(a_n=\dfrac(n+1)!b_n\). Then \[a_n=(n+1)!b_n~~\implies~~a_{n+1}=(n+2)!\,b_{n+1}\] So, \[\begin{align*} a_{n+1}&=\frac{n+2}{n}a_n\\ (n+2)!\,b_{n+1}&=\frac{(n+2)!}{n}b_n\\ b_{n+1}&=\frac{1}{n}b_n \end{align*}\] Next set \(b_n=\dfrac{1}{(n-1)!}c_n\). \[b_n=\frac{1}{(n-1)!}c_n~~\implies~~b_{n+1}=\frac{1}{n!}c_{n+1}\] We have \[\begin{align*} b_{n+1}&=\frac{1}{n}b_n\\ \frac{1}{n!}c_{n+1}&=\frac{1}{n}\times \frac{1}{(n-1)!}c_n\\ c_{n+1}&=c_n\\ c_{n+1}-c_n&=0 \end{align*}\] If we were to sum over \(n=1\) to \(n=k-1\), we'd have \[\sum_{n=1}^{k-1}(c_{n+1}-c_n)=0\] which is telescoping. The sum reduces to \(c_k-c_1=0\), or \(c_k=c_1\). This will be our closed form. We have that \(a_n=\dfrac{(n+1)!}{(n-1)!}c_n=n(n+1)c_n\). Given that \(a_1=1\), we find \[1=\dfrac{2!}{0!}c_1~~\implies~~c_1=\frac{1}{2}\] which gives the closed form \[a_n=\frac{n(n+1)}{2}\]
anonymous
  • anonymous
Another approach, just because :) \[a_{n+1}=\frac{n+2}{n}a_n=a_n+\frac{2}{n}a_n\] Let \(F(x)=\sum\limits_{n\ge1}\dfrac{a_n}{n}x^n\) denote the generating function for \(\dfrac{1}{n}a_n\). We have \[F'(x)=\sum_{n\ge1}a_nx^{n-1}~~\implies~~xF'(x)=\sum_{n\ge1}a_nx^n\] Now, \[\begin{align*} a_{n+1}&=a_n+\frac{2}{n}a_n\\\\ \sum_{n\ge1}a_{n+1}x^n&=\sum_{n\ge1}a_nx^n+2\sum_{n\ge1}\frac{a_n}{n}x^n\\\\ \sum_{n\ge1}a_{n+1}x^{n+1}&=x\sum_{n\ge1}a_nx^n+2x\sum_{n\ge1}\frac{a_n}{n}x^n\\\\ a_1x+\sum_{n\ge1}a_{n+1}x^{n+1}&=x\sum_{n\ge1}a_nx^n+2x\sum_{n\ge1}\frac{a_n}{n}x^n+a_1x\\\\ \sum_{n\ge0}a_{n+1}x^{n+1}&=x^2F'(x)+2xF(x)+x\\\\ \sum_{n\ge1}a_nx^n&=x^2F'(x)+2xF(x)+x\\\\ xF'(x)&=x^2F'(x)+2xF(x)+x\\\\ (1-x)F'(x)-2F(x)&=1\\\\ F'(x)-\frac{2}{1-x}F(x)&=\frac{1}{1-x}\end{align*}\] We have \[F(x)=\exp\left(-2\int\frac{dx}{1-x}\right)=\exp(2\ln|1-x|)=(1-x)^2\] \[\begin{align*} (1-x)^2F'(x)-2(1-x)F(x)&=1-x\\\\ \frac{d}{dx}\left[(1-x)^2F(x)\right]&=1-x\\\\ (1-x)^2F(x)&=x-\frac{1}{2}x^2+C&C=0\text{ since }F(0)=0\\\\ F(x)&=\frac{2x-x^2}{(1-x)^2}\end{align*}\]
anonymous
  • anonymous
Oops, the generating function should be \(F(x)=\dfrac{2x-x^2}{\color{red}2(1-x)^2}\).
Loser66
  • Loser66
Wowwwwwwwwww. I need study harder!!!

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