Suppose that a geometric progression (GP) $1, q, q^{2}, \ldots$ (where $|q|<1)$ is to be constructed such that every term of this GP is a constant multiple (say $k$ ) of sum of all the subsequent terms. For what values of $k$ is this problem solvable?
My Approach: $1+q+q^{2}+\cdots \cdot \alpha$
$q=k\left(q^{2}+q^{3}+\cdots\right)$
$q=k \times \frac{q^{2}}{(1-q)}$
$k q=1-q$
$S_{0} \quad k=\frac{1-q}{q}$
Am I Correct?
Yes, it seems correct. But I think it is better to use a general exponent $n$. Let $$s_n=\sum_{i=0}^{n}q^i=\frac{1-q^{n+1}}{1-q}.$$ And let $S=\lim_{n \to \infty}s_n=\frac{1}{1-q}$. Now, we require that $\forall n \in \mathbb{N}$, $$q^n=k(q^{n+1}+q^{n+2}+...)$$ In other terms, $$k=\frac{q^n}{S-s_n}$$ $$k=\frac{q^n}{\frac{1}{1-q}-\frac{1-q^{n+1}}{1-q}}$$ $$...k=\frac{1-q}{q}.$$