Assume that you have $n$ positive values $C_1,C_2,\ldots,C_n$, and you have $n$ values $g_1,g_2,\ldots,g_n$ where each $g_t\in[0,0.1]$.
Do we then have that
$$\frac{\sum\limits_{t=1}^n\frac{C_tg_t}{(1+g_t)^t}}{\sum\limits_{t=1}^n\frac{C_t}{(1+g_t)^t}}\ge\frac{\sum\limits_{t=2}^n\frac{C_tg_t}{(1+g_t)^{t-1}}}{\sum\limits_{t=2}^n\frac{C_t}{(1+g_t)^{t-1}}}?$$
As a counterexample, if $$ \left\lbrace \begin{align*} &n=2\\[4pt] &C_1=C_2=1\\[4pt] &g_1,g_2=\frac{1}{20},\frac{1}{10}\\[4pt] \end{align*} \right. $$ then letting $L,R$ denote respectively the $\text{LHS},\text{RHS}$ of your proposed inequality, we get $$ \left\lbrace \begin{align*} R&=\frac{1}{10}\\[4pt] L&=\frac{331}{4520}\\[4pt] R-L&=\frac{121}{4520}\\[4pt] \end{align*} \right. $$ so $R > L$.
In fact, for the case $n=2$, your proposed inequality holds if and only if $g_1 \ge g_2$.