∀n∈N, 1+2x+3x2+ +n xn-1= (x∈R,x≠1)1)1-(n+1)xn+n xn+1(1-x)22)(n+1)xn(1-x)23)1-(n+1)xn+n xn+1(1+x)24)(n-1)xn(1+x)2 - Maths - Principle of Mathematical Induction

Question

∀ n ∈ N ,   1 + 2 x + 3 x 2 + + n x n - 1 =  
( x ∈ R , x ≠ 1 ) 1 ) 1 - ( n + 1 ) x n + n x n + 1 ( 1
- x ) 2 2 ) ( n + 1 ) x n ( 1 - x ) 2 3 ) 1 - ( n + 1 ) x n + n x n
+ 1 ( 1 + x ) 2 4 ) ( n - 1 ) x n ( 1 + x ) 2

Brijendra Pal · Accepted Answer

Hi,

∀ n ∈ N , 1 + 2 x + 3 x 2 + . . . . + n . x n - 1 = ( x ∈ R , x ≠ 1 ) 1 ) 1 - ( n + 1 ) x n + n . x n + 1 ( 1 - x ) 2 2 ) ( n + 1 ) x n ( 1 - x ) 2 3 ) 1 - ( n + 1 ) x n + n . x n + 1 ( 1 + x ) 2 4 ) ( n - 1 ) x n ( 1 + x ) 2

$\forall n \in N, 1 + 2 x + 3 x^{2} + . . . . + n . x^{n - 1} = (x \in R, x \neq 1) 1) \frac{1 - (n + 1) x^{n} + n . x^{n + 1}}{(1 - x)^{2}} 2) \frac{(n + 1) x^{n}}{(1 - x)^{2}} 3) \frac{1 - (n + 1) x^{n} + n . x^{n + 1}}{(1 + x)^{2}} 4) \frac{(n - 1) x^{n}}{(1 + x)^{2}}$