Find the constant term in the expansion of 3x 1x34 Show tha

Find the constant term in the expansion of (3x - 1/x^3)^4 Show that (3 + 3) is a divisor of (n +1)! + n! - (n - 1)!

Solution

2)(a).The formula is (a+b)⁴=a⁴+4a³b+6a²b²+4ab³+b^4.,

^Here. a= 3x, b= -1/x^3.,

  now the constant term of the given problem is givenby the second term of the formula...

ie, 4a³b.... So 4a³b=4(3x)³(-1/x³)

=4(27)(x³)(-1/x³)

=-108

More over if we use the above formula to expand the given problem we get 81x⁴-108+54/x⁴-12/x⁸..... This rsult lelead to te answer., the constant term is = -108

2.(b).let (n+1)! + n! -3(n-1)! =(n-1)! *n *(n+1) +(n-1)! *n -3(n-1)!

=n-1)! [n(n+1)+n-3]

=(n-1)! [ n²+2n-3 ]

=(n-1)! [(n-1)(n+3)]

The above expression has a factor (n+3)... There fore it must divided by (n+3).... Hence the proof.