Using the relation A.M.$\ge$G.M., prove that $\left(a+b\right)·\left(b+c\right)·\left(c+a\right)>abc;$ if $a, b, c$ are positive real numbers.

Find the sum to $n$-terms of the sequence.

$1+5+13+29+61+\dots \dots$ up to $n$ terms

Find the sum to $n$ terms of the sequence $3+33+333+3333+\dots \dots$ up to $n$ terms.

Find the sum to $n$-terms of the sequence

$\frac{1}{1·3·5}+\frac{1}{3·5·7}+\frac{1}{5·7·9}+\dots$

Find the sum to $n$-terms of the sequence

$1·3·2^2+2·4·3^2+3·5·4^2+\dots$