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!!