The first $n$ terms of a series are given by \({{\log }_{a}}3+{{\log }_{a}}27+{{\log }_{a}}243+\ld

Question

The first $n$ terms of a series are given by ${{\log }_{a}}3+{{\log }_{a}}27+{{\log }_{a}}243+\ldots +{{\log }_{a}}{{3}^{2n-1}}$, where $a$ is a positive constant. Show that the series is an arithmetic series.[2] Given that sum of the first $30$ terms of the series is $300$, find the value of $a$.[2] A geometric series has first term $c$ and common ratio $r$, where $c$ and $r$ are non-zero. An arithmetic series has first term $b$ and common difference $d$, where $b$ and $d$ are non-zero. It is given that the ${{5}^{\mathrm{th}}}$, ${{8}^{\mathrm{th}}}$ and ${{10}^{\mathrm{th}}}$ terms of the arithmetic series are equal to the 2nd, 3rd and 4th of the geometric series respectively. Show that $r$ satisfies the equation $3{{r}^{2}}-5r+2=0$ and hence find the sum to infinity in terms of $c$.[4]

Accepted Answer

(a)(i) Arithmetic series with common difference $\log_a9$ (a)(ii) $a=27$ (b) $3r^2-5r+2=0$, $r=\dfrac23$, and $S_\infty=3c$

2023 EJC Promo Q6

2023 EJC Promo Q6

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