MCQ
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
  • $\{ {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 $G.P$

Answer

Correct option: A.
$\{ {t_n}\} $ is an $A.P.$, $\{ {s_n}\} $ is a $G.P.$
a
(a) ${t_n} = \log \left( {\frac{{{5^{n + 1}}}}{{{3^{n - 1}}}}} \right)$ ; ${s_n} = \left[ {\log (5/3)} \right]{\,^n}$

${t_1} = \log 25$; ${s_1} = {\left[ {\log \,\,5/3} \right]^1}$

${t_2} = \log \frac{{125}}{3}$; ${s_2} = {\left[ {\log \,5/3} \right]^2}$

${t_3} = \log \frac{{625}}{9}$; ${s_3} = {\left[ {\log \,5/3} \right]^3}$

Clearly ${t_n}$ is an $A.P.$ and ${s_n}$ is $G.P.$

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