Binomial Problem- 10 November

Question

The power of !!x!! with largest coefficient !!sqrt3(x + 1/sqrt3)^20!!

!!a)" "12!!

!!b)" "13!!

!!c)" "14!!

!!d)!!None

Solution

In order to calculate the power of !!x!! with largest coefficient we need to calculate largest term.

Method 1

!!sqrt3(x + 1/sqrt3)^20!!

For !!(x+a)^n!!

!!(T_(r+1)/T_r) = ((nC_r)(x^(n-r))(a^r))/((nC_(r-1))(x^(n-r+1))(a^(r-1))!!

!!(T_(r+1)/T_r) = ((20C_r)/(20C_(r-1))xx(a/x))!!

!!(T_(r+1)/T_r) = ((20C_r)/(20C_(r-1))xx(1/(sqrt3x))!!

!!(T_(r+1)/T_r) = ((21-r)/r)xx(1/(sqrt3x))!!

!!(T_(r+1)/T_r)!! should be greater than !!1!!

!!(T_(r+1)/T_r) >= 1!!

!!=> ((21-r)/r)xx(1/sqrt3) >=1!!

!!=> r <= 7.69!!

Hence !!T_(r+1) = 8!! is greatest term.

This will give coefficient of !!x = 13!!

Method 2

Taking !!x = 1!! as !!sqrt3(1 + 1/sqrt3)^20!!.

In equation !!(1 + y)^n!!, largest term can be calculated as;

!!m = |(y(n+1))/(y+1)|!!

!!=> m = |((1/sqrt3)(20+1))/(1 + (1/sqrt3))|!!

!!=> m = 7.69!!

Largest term is greatest integer of !!m!! plus 1!!

Hence largest term comes !!T_8!!

This will give largest coefficient of !!x = 13!!