A community for students.
Here's the question you clicked on:
 0 viewing
butterflydreamer
 one year ago
help please :)  BINOMIAL THEOREM CC's question below:
butterflydreamer
 one year ago
help please :)  BINOMIAL THEOREM CC's question below:

This Question is Closed

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1Prove: \[1 + \left(\begin{matrix}10 \\ 2\end{matrix}\right)3^{2} + \left(\begin{matrix}10 \\ 4\end{matrix}\right)3^{4} + \left(\begin{matrix}10 \\ 6\end{matrix}\right)3^{6} + \left(\begin{matrix}10 \\ 8\end{matrix}\right)3^{8} + 3^{10} = 2^9 (2^{10} +1)\]

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1I'm considering: \[(1+x)^{10} = \left(\begin{matrix}10 \\ 0\end{matrix}\right) +\left(\begin{matrix}10 \\ 1\end{matrix}\right)x + \left(\begin{matrix}10 \\ 2\end{matrix}\right)x^2 + \left(\begin{matrix}10 \\ 3\end{matrix}\right)x^3 + ... + x^{10} \] But i have no idea where to go from here. If i look at the RHS, then i came up with something....?: \[2^9 ( 2^{10} + 1) \rightarrow 2^{n1} (2^{n} +1) \] But i'm not how to get there :/ I also noticed on the LHS, the things are all even. Idk .

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3``` \left(\begin{matrix}10 \\ 2\end{matrix}\right) ``` is better writen as ``` \binom{10}{2} ``` \[\binom{10}{2}\]

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1oh, thanks for the tip LOL

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3(x1)^(10) = cx^10 cx^9 +cx^8 cx^7.. right?

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3what is: (x+1)^(10) + (x1)^(10)

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3\[2 (x^{10}+45 x^8+210 x^6+210 x^4+45 x^2+1)\]

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1ohhh true true . That makes sense! So that's how they got the even parts only.

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3im an idiot, so i just use whatever comes to mind :)

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3oh im not sure how THEY did it .. but this might be a start

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1yeess. LOL i can't believe i expanded wrong xD oh gosh. Okay sooo, what should we do next?

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3dunno offhand, i was just trying to get your thought to get rid of the odd parts

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3since LHS and RHS are finite numbers, cant we just let x=3 and work the math to prove?

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3(x+1)^(10) + (x1)^(10) 4^(10) + 2^(10) 2^(10) 2^10 + 2^(10) 2^(10) [2^(10)+1]

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3and divide by 2, since we are twice as heavy

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3or something like that

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1.... oh... o.o

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1can we solve the question like that though? LOL. I've been trying to sub all kinds of weird numbers everywhere and trying to integrate things . I've been getting nowhere tbh LOl

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3direct solutions tend to work out great to me ... \[1 + \left(\begin{matrix}10 \\ 2\end{matrix}\right)x^{2} + \left(\begin{matrix}10 \\ 4\end{matrix}\right)x^{4} + \left(\begin{matrix}10 \\ 6\end{matrix}\right)x^{6} + \left(\begin{matrix}10 \\ 8\end{matrix}\right)x^{8} + x^{10} =\frac{(x+1)^{10}+(x1)^{10}}{2}\]

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3when x=3, ive already worked the numerator to 2^(10) [2^(10)+1] right?

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1hmm okay then. Thank you! It's just that for the prev. questions i've been working on, i've had to use a range of differentiation/integration and substitution methods so i thought this question would've involved similar steps but thanks again! Your method was way simpler.

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3\[1 + \left(\begin{matrix}10 \\ 2\end{matrix}\right)3^{2} + \left(\begin{matrix}10 \\ 4\end{matrix}\right)3^{4} + \left(\begin{matrix}10 \\ 6\end{matrix}\right)3^{6} + \left(\begin{matrix}10 \\ 8\end{matrix}\right)3^{8} + 3^{10}\] \[\frac{(3+1)^{10}+(31)^{10}}{2}\] \[\frac{4^{10}+2^{10}}{2}\] \[\frac{2^{10}(2^{10}+1)}{2}\] \[{2^{9}(2^{10}+1)}\]

amistre64
 one year ago
Best ResponseYou've already chosen the best response.3well, it was your way, i just built on it :)

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1aahh okay! Thanks again :D!
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.