`3C_0+3^2(C_1)/2+3^3(C_2)/3+.............3^(n+1)*(C_n)/(n+1)` eqaul to

Explanation

`because (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + ...+ C_(n) x^(n) `...(i)
Integrating on both sides of Eq. (i) within limits 0 to , we get
`int_(0)^(3)(1 + x)^(n) dx = int_(0)^(3)(C_(0) + C_(1)x + C_(2)x^(2) + C_(3) x^(3) + ...+ C_(n) x^(n)) dx `
`rArr [((1+ x)^(n+1))/(n+1)]_(0)^(3) = [C_(0) x+ (C_(1) x^(2))/(2) + (C_(2) x^(3))/(3) + (C_(3)x^4)/(4) + ...+ (C_(n) x^(n+1))/(n+1)]_(0)^(3)`
`rArr (4^(n+1)-1)/(n+1) = 3C_(0) + (3^(2)C_(1))/(2) + (3^(3)C_(2))/(3) + (3^(4)C_(3))/(4) + ...+ (3^(n+1)C_(n))/(n+1)`
Hence ,
`3C_(0) + (3^(2)C_(1))/(2) + (3^(3) C_(2))/(3) + (3^(4)C_(3))/(4) + ...+ (3^(n+1) C_(n))/(n+1) = (4^(n+1)-1)/(n+1)`
I.Aliter
`LHS = 3C_(0) + (3^(2)C_(1))/(2) + (3^(2) C_(2))/(3) + (3^(4) C_(3))/(4) + ...+ (3^(n+1)C_(n))/(n+1) `
`= 3.1 + (3^(2) *n)/(2) + (3^(3).(n-1))/(1*2*3) + (3^(4) .n(n-1)(n-2))/((1*2*3*4) ) + ...+ (3^(n+1))/(n+1)`

`= (1)/((n+1))[3*(n+1)+(3^(2)(n+1)n)/(1*2) + (3^(3) (n+1) n(n-1))/(1*2*3) + (3^(4) (n+1)n(n-1)(n-2))/(1*2*3*4) + ..+ 3^(n+1)]`
Put n + 1 = N , then
`LHS= (1)/(N) [3N + (3^(2) N(N-1))/(2!) + (3^(3)N(N-1)(N-2))/(3!)+ (3^(3)N(N-1)(N-2)(N-3))/(4!)+...+ 3^(N)]`
`= (1)/(N)[""^(N)C_(1) (3) + ""^(N)C_(2) (3)^(2) + ""^(N)C_(3) (3)^(3) + ...+ ""^(N)C_(N) (3)^(N) ]`
` = (1)/(N)[""^(N)C_(0) + ""^(N)C_(1)(3) + ""^(N)C_(2)(3)^(2) + ""^(N)C_(3) (3)^(3) + ...+ ""^(N)C_(n) (3)^(N) - ""^(N)C_(0)]`
` = (1)/(N){(1+3)^(N)-1} = (4^(N) -1)/(N) = (4^(n+1)-1)/(n+1) = RHS `
`LHS = 3C_(0) + 3^(2) (C_(1))/(2) + (3^(3) C_(2))/(3) + (3^(4)C_(3))/(4) + ...+ (3^(n+1)C_(n))/(n+1)`
`= sum_(r=0)^(n) (3^(r+1)*""^(n)C_(r))/((r+1))= sum_(r=0)^(n) (3^(r+1)*""^(n+1)C_(r+1))/((n+1)) " "[because (""^(n+1)C_(r+1))/(n+1) = (""^(n)C_(r))/(r+1)]`
`= (1)/((n+1)) sum_(r=0) ^(n) ""^(n+1)C_(r+1) *3^(r+1)`
`= (1)/((n+1)) (""^(n+1)C_(1) *3+""^(n+1)C_(2)*3^(2) + ""^(n+1)C_(3) *3^(3) + ""^(n+1)C_(n+1) *3^(n+1))`
`= (1)/((n+1)) [(1 + 3)^(n+1" - " n+1)C_(0)]`
` = (4^(n+1)-1)/(n+1) = RHS `