Question

Find the sum of infinite series :
1+(1+b)r+(1+b+b^2)r^2 + (1+b+b^2+b^3)r^3+......
where r and b are proper fractions.

Answer 1

Let the sum be S
\[S=1+ (1+b)r+(1+b+b^2)r^2+(1+b+b^2+b^3)r^3+...\]
multiply S by r, we get
\[rS=r+(1+b)r^2+(1+b+b^2)r^2+(1+b+b^2)r^3+....\]
now subtract S by rS we get
\[S-rS=1+br+b^2r^2+b^3r^3+...\]
or
\[S(1-r)=1+br+b^2r^2+b^3r^3\]
could you solve for s now, the right side is just GP

Answer 2

@shubham.bagrecha do you get this?

Answer 3

yes