## butterflydreamer one year ago BINOMIAL THEOREM question: http://prntscr.com/8ib6g8 Find the numerically greatest coefficient in the expansion of (2 - x^2/4)^10.

1. butterflydreamer

I'm confused on what to do with the negative sign?? I know we use the formula: $\frac{ T _{k+1} }{ T_{k}} = \frac{ n - k + 1 }{ k } \times \frac{ b }{ a }$

2. butterflydreamer

Or do we ignore the negative? O_O

3. dan815

2^10

4. Empty

I don't know what this is, so I don't know what they mean. I think the binomial is $(a+b)^n = \sum_{k=0}^n \binom{n}{k}a^kb^{n-k}$ So I'm not so sure what they're wanting here, I am guessing coefficient on x after it's all expanded out right?

5. butterflydreamer

This is what i did but i'm not sure if i'm solving this correctly :S http://prntscr.com/8ib6g8

6. dan815

it would be -1280

7. dan815

if negatives dont matter then that is right i think

8. zepdrix

I know this doesn't really help with the problem all that much, but I would factor the 1/4 out of each term, that is just getting in the way in my opinion,$\large\rm \left(2-\frac{x^2}{4}\right)^{10}=\frac{1}{4^{10}}\left(8-x^2\right)^{10}$

9. zepdrix

Yah I would imagine that's what they meant by "numerically greatest", like ignore the sign :O that sounds right.

10. dan815

sheesh why cant they just say magnitude or abs value -.-

11. butterflydreamer

hmm i'm lost xD I don't know whether it will be -1280 or 1280... I think maybe it'll be -1280??

12. zepdrix

why x=1? :o does that just make the rest of the calculations easier or something?

13. butterflydreamer

i actually have no idea LOLL. I just got taught to set x = 1 if there was no given value of x? So unless the question gave a specific value of x, you'd just plug in x = 1 :/ If they asked for the term independent of x, then you set x = 0

14. zepdrix

I don't know that weird T formula 0_o I'mma do it the long way and see if that checks out.$\large\rm \frac{T_{k+1}}{T_k}=\frac{\color{orangered}{\left(\begin{matrix}10 \\ k+1\end{matrix}\right)}2^{10-(k+1)}\left(-\frac{x}{4}\right)^{k+1}}{\color{orangered}{\left(\begin{matrix}10 \\ k\end{matrix}\right)}2^{10-k}\left(-\frac{x}{4}\right)^{k}}$Which I guess simplifies down a bit right? Ummm$\large\rm \frac{T_{k+1}}{T_k}=\frac{\color{orangered}{\frac{10!}{(k+1)!(10-(k+1))!}}\left(-\frac{x}{4}\right)^1}{\color{orangered}{\frac{10!}{k!(10-k)!}}2^1}$And more simplifying -_-$\large\rm \frac{T_{k+1}}{T_k}=\frac{k!(10-k)!}{(k+1)!(10-(k+1))!}\cdot\left(-\frac{x}{8}\right)$and further -_-$\large\rm \frac{T_{k+1}}{T_k}=\frac{(10-k)}{(k+1)}\cdot\left(-\frac{x}{8}\right)$Hmmmm yah I'm seeing how you got 11 :O Thinking...

15. dan815

1280 i read some post where numerical greatest means the magnitude of the number

16. zepdrix

Not seeing how you got 11* blah

17. dan815

pls show me appreciation with your medals

18. butterflydreamer

is there even a difference between "greatest coefficient" and "numerically greatest coefficient" ? o.o But alrighties, i guess i'll stick with 1280...

19. zepdrix

Oh your formula works actually,$\large\rm \frac{T_{k+1}}{T_k}=\frac{(10-\color{royalblue}{(k)})}{(\color{royalblue}{k}+1)}\cdot\left(-\frac{x}{8}\right)$ $\large\rm \frac{T_{k+1}}{T_k}=\frac{(10-\color{royalblue}{(k-1)})}{(\color{royalblue}{k-1}+1)}\cdot\left(-\frac{x}{8}\right)$ $\large\rm \frac{T_{k+1}}{T_k}=\frac{10-k+1}{k}\cdot\left(-\frac{x}{8}\right)$Ok I'll simmer down XD

20. dan815

yes greatest coefficient means the biggest number where positive numbers are greater than the negative numbers Numerically greatest means the number that is farthest away from 0 or the origin

21. butterflydreamer

LMAO Zepdrix xD Your dedication is so strong. Binomials give me headaches. Dan, wait, so if it is the number farthest away from 0 or the origin, wouldn't it be -1280?

22. dan815

yes thats right

23. butterflydreamer

okaaayyy ^_^ Thank you ALL~ <3.

24. dan815

no problem

25. dan815

noob

26. butterflydreamer

-.-

27. zepdrix

ooo snap

28. anonymous

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29. Jhannybean

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