(i) (1\cdot2+3\cdot4+5\cdot8+7\cdot16+\cdots)

(t_n=(2n-1)\cdot 2^n)

[ S_n=\sum_{k=1}^{n}(2k-1)2^k ] Using standard AG sum method: (S=1\cdot2+3\cdot4+5\cdot8+\cdots+(2n-1)2^n)

(ii) (2\cdot3+4\cdot3^2+6\cdot3^3+8\cdot3^4+\cdots)

(t_n=2n\cdot 3^n)

[ S_n=\sum 2k\cdot 3^k = 2\sum k\cdot 3^k ] Known formula (\sum_{k=1}^{n}kx^k = x\dfrac{1-(n+1)x^n+nx^{n+1}}{(1-x)^2})

[ S_n=2\cdot 3\cdot\dfrac{1-(n+1)3^n+n\cdot3^{n+1}}{(1-3)^2}=\dfrac{3}{2}\bigl[1-(n+1)3^n+n\cdot3^{n+1}\bigr] ]

(iii)–(v) Similar AG series; apply (S-rS) method.

Answers obtained via AG summation formula.