Two sequences $\{ {t_n}\} $ and $\{ {s_n}\} $ are defined by ${t_n} = \log \left( {\frac{{{5^{n + 1}}}}{{{3^{n - 1}}}}} \right)\,,\,\,{s_n} = {\left[ {\log \left( {\frac{5}{3}} \right)} \right]^n}$, then
Two sequences $\{ {t_n}\} $ and $\{ {s_n}\} $ are defined by ${t_n} = \log \left( {\frac{{{5^{n + 1}}}}{{{3^{n - 1}}}}} \right)\,,\,\,{s_n} = {\left[ {\log \left( {\frac{5}{3}} \right)} \right]^n}$, then
- A
$\{ {t_n}\} $ is an $A.P.$, $\{ {s_n}\} $ is a $G.P.$
- B
$\left\{ {{t_n}} \right\}$ and $\{ {s_n}\} $ are both $G.P.$
- C
$\{ {t_n}\} $ and $\{ {s_n}\} $are both $A.P.$
- D
$\left\{ {{s_n}} \right\}$ is a $G.P.$, $\left\{ {{t_n}} \right\}$ is neither $A.P.$ nor $G.P$
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Three numbers are in an increasing geometric progression with common ratio $\mathrm{r}$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $\mathrm{d}$. If the fourth term of GP is $3 \mathrm{r}^{2}$, then $\mathrm{r}^{2}-\mathrm{d}$ is equal to:
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