If `(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + … + C_(n) x^(n)`,
Show that `(2^(2))/(1*2) C_(0) + (2^(3))/(2*3) C_(1) + (2^(4))/(3*4)C_(2) + ...+ (2^(n+2)C_n)/((n+1)(n+2))= (3^(n+2)-2n-5)/((n+1)(n+2))`

Explanation

Given,
Integrating on both sides of Eq. (i) within limits 0 to 3, 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)^(x)`
` = [C_(0) x + (C_(1) x^(2))/(2) + (C_(2)x^(3))/(3) + ...+ (C_(n) x^(n+1))/(n+1)]_(0)^(x)`
`rArr ((1 +x)^(n+1)-1)/((n+1))= C_(0) x + (C_(1) x^(2))/(2) + (C_(1) x^(3))/(3) + ...+ (C_(n) x^(n+1))/(n+1) ` ...(ii)
Again, integrating both sides of Eq.(ii) within limits 0 to 2, we get
`int_(0)^(2) (C_(0) x + (C_(1)x^(2))/(2) + (C_(2) x^(3))/( 3) + ...+ (C_(n) x^(n+1))/(n+1))dx `
` rArr (1)/((n+1)) (((1+ x)^(n+1))/(n+2)- x)]_(0)^(2) = [(C_(0)x^(2))/(1*2) + (C_(1)x^(3))/(2*3) + (C_(2)x^(4))/(3*4)+ ...+ (C_(n) x^(n+2))/((n+1)(n+2))]_(0)^(2)`
`rArr (1)/((n+1)) {(3^(n+2))/(n+2) -2-(1)/(n+2)} = (2^(2))/(1*2) C_(0) + (2^(3))/(2*3) + C_(1) (2^(4))/(3*4) C_(2)+ ...+ (2^(n+2)C_(n))/((n+1)(n+2))`
Hence , `(2^(2))/(1*2)C_(0) + (2^(3))/(2*3) + C_(1) + (2^(4))/(3*4)C_(2) + ...+ (2^(n+2)C_(n))/((n+1)(n+2))= (3^(n+2) -2n-5)/((n+1)(n+2))`
I.Aliter
`LHS = (2^(2))/(1*2)C_(0) + (2^(3))/(2*3)C_(1) + (2^(4))/(3*4) + C_(2) + ...+ (2^(n+2)C_(n))/((n+1)(n+2))`
` = (2^(2))/(1*2)(1) + (2^(3))/(2*3)n + (2^(4))/(3*4) + (n(n-1))/(1*2) + ...+ (2^(n+2).1)/((n+1)(n+2))`
`(1)/((n+1)(n+2)) {((n+2)(n+1))/(1*2)2^(2) + ((n+2)(n+1)n)/(1*2*3) 2^(3) + ((n+2)(n+1)n(n-1))/(1*2*3*4) 2^(4) + ...+ 2^(n+2)}`
Put n+2 = N , then we get
`= (1)/(N(N-1)){(N(N-1))/(1*2) 2^(2) + (N(N-1)(N-2))/(1*2*3)+ (N(N-1)(N-2)(N-3))/(1*2*3*4) 2^(4) + ... + 2^(N)}`
`= (1)/(N(N-1)){""^(N)C_(2)(2)^(2) + ""^(N)C_(3)(2)^(3) + ""^(N)C_(4) + (2)^(4) + ...+ ""^(N)C_(N) (2)^(N)]`
`= (1)/(N(N-1)){""^(N)C_(0) + ""^(N)C_(1)(2) + ""^(N)C_(4) + (2)^(2) + ""^(N)C_(3)(2)^(3) + ""^(N)C_(4) (2)^(4) + ...+ ""^(N)C_(N)- (2)^(N)-""^(N)C_(0) - ""^(N)C_(1)(2)]`
`= (1)/(N(N-1)) {(1+ 2)^(N) -1-2N}`
`= (3^(n+2)-1-2(n+2))/((n+2)(n+1)) = (3^(n+2) - 2n-5)/((n+1)(n+2))= RHS `
II.Aliter
LHS = `(2^(2))/(1*2) *C_(0) + (2^(3))/(2*3) *C_(1) + (2^(4))/(3*4) *C_(2) + ...+ (2^(n+2)*C_(n))/((n+1)(n+2))`
` = sum_(r=1)^(n+1) (2^(r+1))/(r +1)*""^(n)C_(r-1)`
` = sum_(r=1)^(n+1) (2^(r+1)*""^(n+2)C_(r+1))/((n+1)(n +2))[because (""^(n+2)C_(r+1))/((n+1)(n+2)) = (""^(n)C_(r-1))/(r(r+1))]`
`= (1)/((n+1)(n+2) )sum_(r=1)^(n+1) ""^(n+2)C_(r+1) *2^(r+1)`
`= (1)/((n + 1)(n+2))[""^(n+1)C_(2) *2^(2) + ""^(n +2)C_(3)*2^(3) +...+ ""^(n+2)C_(n+2) *2^(n+2)]`
`= (1)/((n + 1)(n+2))[(1 + 2)^(n+2" - "n +2)C_(0) - ""^(n+2)C_(1) *2^(1)]`
` = ((3^(n+2) - 2n-5))/((n+1)(n+2)) = RHS ` .