Equating Coefficients (1 Viewer)

[DJel]

Question: "By equating the coefficients of (1+x)¹⁰ and (1+x)(1+x)⁹, show that ¹⁰C₄ = ⁹C₄ + ⁹C₃"

I understand how to get the coefficients of (1+x)¹⁰ and from that I can show the relationship using factorial notation. However, I'm unsure what to do with "(1+x)(1+x)⁹".

Thanks,

DJel.

 

[tacogym27101990]

Question: "By equating the coefficients of (1+x)¹⁰ and (1+x)(1+x)⁹, show that ¹⁰C₄ = ⁹C₄ + ⁹C₃"

I understand how to get the coefficients of (1+x)¹⁰ and from that I can show the relationship using factorial notation. However, I'm unsure what to do with "(1+x)(1+x)⁹".

Thanks,

DJel.

you expand the rhs binomally aswell
and u get something like this
(10c0+10c1x+...+10c10x^10)=(1+x)(9c0 + 9c1x+ 9c2x^2+.........+ 9c9x9)
then equate the coefficients of x^4
so you get LHS=10c4x^4
and RHS= 1.9c4x^4 + x(9c3x^3)
= 9c4x^4 + 9c3x^4
hence 10c4= 9c4+9c3

 

[DJel]

Thanks for that solution.

Also, I am having a bit of trouble with the short way of finding the greatest coefficient. I know how to expand them out and find the greatest coefficient, however, this just wastes time. What would be the quickest way of doing a question like so;

"Find the greatest coefficient of (2x² - 3/x)¹¹"?

DJel.

 

[lyounamu]

Thanks for that solution.

Also, I am having a bit of trouble with the short way of finding the greatest coefficient. I know how to expand them out and find the greatest coefficient, however, this just wastes time. What would be the quickest way of doing a question like so;

"Find the greatest coefficient of (2x² - 3/x)¹¹"?

DJel.

There is no quickest way, unfortunately. Expansion will be the best way as well.

 

[Aerath]

Really? I thought you could use T_r+1 = a^n-r*b^r to find T_r+1 and T_r.
Then solve the coeff of T_r+1 / T_r > 1

And after that, sub it in to find the T with the greatest coefficient.

 

[lyounamu]

Really? I thought you could use T_r+1 = a^n-r*b^r to find T_r+1 and T_r.
Then solve the coeff of T_r+1 / T_r > 1

And after that, sub it in to find the T with the greatest coefficient.

That's the method he was talking about wasn't it?

I call that expansion method. lol
(because you expand it and simplify to get the right term).