Binomial question D: (1 Viewer)

[youngsky]

"When (3+2x)^n (n is a positive integer) is written out as a polynomial in x, the coefficients of x^5 and x^6 have the same value."

i. Find the value of n.
ii. Show that t(k+1)/t(k) = 2(15-k)/3k
iii. Hence or otherwise, show that the coefficients of x^5 and x^6 are the largest coefficients in the expansion.

I'm mainly having trouble with part i, help is appreciated, thanks

 

[asianese]

What is the coefficient of x^5 and x^6?



Which value of k will give a power of x^5 (and hence x^6?)

 

[deltaforce22]

What is the coefficient of x^5 and x^6?



Which value of k will give a power of x^5 (and hence x^6?)

How did you arrive to that deduction?

 

[youngsky]

Yeah i got past that part, k = 5, k = 6, but the problem is simplifying the coefficients' expression

I ended up with something like nC5 x 3^-5 = 2 x nC6 x 3^-6

@deltaforce its the general expression for binomial expansions

 

[omgiloverice]

How did you arrive to that deduction?

What do you mean?

 

[deltaforce22]

I see. Well let k=5 then k=6 and since they possess an equivalent coefficient, through common-sense we equate them and find n.

 

[omgiloverice]

Yeah i got past that part, k = 5, k = 6, but the problem is simplifying the coefficients' expression

I ended up with something like nC5 x 3^-5 = 2 x nC6 x 3^-6

@deltaforce its the general expression for binomial expansions

You're correct keep going

 

[youngsky]

lol nice edit.

after that i just expanded the nC5 and nC6 into n!/5!(n-5)! and n!/6!(n-6)! but that's about it

help?

 

[deltaforce22]

Wait. Isn't asianese missing an 'nCk' in his thing?

 

[asianese]

Sorry - apologies. Fixed now.

 

[youngsky]

never mind - i got it now, thanks asianese

n = 14

 

[deltaforce22]

Sorry - apologies. Fixed now.

The man.