1+2/a+3/a^2+4/a^3+...+n/a^n-1

Let S=1+2a+3a

2

+4a

3

+ +na

n−1

Multiply both sides by a, we get

Sa=0+a+2a

2

+3a

3

(n−1)a

n−1

+na

n

Subtract both equations,

S(1−a)=1+a+a

2

+a

3

a

n−1

−na

n

Clearly above series is G.P

Common ratio =a

S(1−a)=

a−1

1(a

n

−1)

−na

n

S=

(a−1)

2

1−(a

n

)

−

1−a

na

n