Question

if 3rd 4th 5th and 6th terms in the expansion of X + y whole power n are respectively a, b, c and d then prove that B square minus AC by c square minus bd equal to 5a by 3c​

Answer 1

Answer:

proved

Step-by-step explanation:

from expansion of (x + y )^n

3rd term = U3

U3 = nc2y²x^n-2 = a

U4 = nc3y³x^n-3 = b

U5 = nc4y^4x^n-4 = c

U6 = nc5y^5x^n-5 = d

now it is required to prove :

b² - ac ÷ c² - bd = 5a ÷ 144c

LHS

=  {nc3y³x^n-3}² - nc2y²x^n-2nc4y^4x^n-4 ÷ {nc4y^4x^n-4}² - nc3y³x^n-3.nc5y^5x^n-5

= y^6.x^2n-6.n².(n-1)²(n-2)(n+1)/12³ ÷ y^8.x^2n-8.n².(n-1)²(n-2)²(n-3)(n+1)/5.12²

= 5x² ÷ 12y²(n-2)(n-3)

RHS = 5a ÷ 144c = 5nc2y²x^n-2 ÷ 144 nc4y^4x^n-4

= 5.n.(n-1)y²x^n-2/2 ÷ 144n.(n-1)(n-2)(n-3)y^4x^n-4/24

= 5.n.(n-1)y²x^n-2/2 ÷ 12n.(n-1)(n-2)(n-3)y^4x^n-4/2

= 5x² ÷ 12y²(n-2)(n-3)

= RHS