(i) (1+(a+b)+(a^2+ab+b^2)+(a^3+a^2b+ab^2+b^3)+\cdots)

(n^{th}) term (= \dfrac{a^n-b^n}{a-b}) (if (a\neq b))

[ S_n = \sum_{k=0}^{n-1}\dfrac{a^k-b^k}{a-b} = \dfrac{1}{a-b}\left[\dfrac{a^n-1}{a-1}-\dfrac{b^n-1}{b-1}\right] ] (assuming (a,b\neq1); special cases handled separately)

(ii) (r+(1+k)r^2+(1+k+k^2)r^3+\cdots)

(n^{th}) term (= \dfrac{1-k^n}{1-k}r^n) (if (k\neq1))

Sum can be obtained by writing (S = \sum r^n\dfrac{1-k^n}{1-k}) and separating into two G.P.s.

Answer: See detailed expansion in solution context for specific (a,b,k).