Plz expain how to solve dis ∀ n∈N ,1+2x+3x2+ +n xn-1=(x∈R, X≠1)1)1-(n+1)xn+n - Maths - Principle of Mathematical Induction

Question

Plz expain how to solve dis ∀   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

Lovina Kansal · Accepted Answer

Dear student
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1-xS=1+x+x2+x3+
+xn-1-nxn1-xS=11-xn1-x-nxn   Here 1+x+x2+x3+
+xn-1 this is forming a GP with a=1 and r=x1=x and ∵Sum of GP=a1-rn1-r 1-xS=1-xn-nxn+nxn+11-xS=1-xn-nxn+nxn+11-x2S=1-xn1+n+nxn+11-x2
Regards

Plz expain how to solve dis... ∀ 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