Question

sum of the series 1+1/3+1/3²+1/3³+...+1/3 power n-1 is a)2/3 b)3/2 c)4/5 d) none of these ​

Answer 1

Answer:

S=1+13+132+133+⋯+13n−1(1)

3S=3(1+13+132+133+⋯+13n−1)

3S=3+1+13+132+133+⋯+13n−2(2)

3S=3+S

2S=3

S=32

This is for infinite sum. But for n terms,

(2)−(1)⟹

2S=3+13n

S=12(3−13n−1)

S=32−12×3n−1

This can also be calculated by observing it as GP with r=13

And using Sn=a(1−rn)1−r

Thanks for this opportunity.