# Test: Geometric Series I - Challenging

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Question 1:   In the geometric series $\text{\hspace{0.17em}}\left({a}_{n}\right)\text{\hspace{0.17em}}$ it’s known that $\text{\hspace{0.17em}}{a}_{10}=2$ . Find product of the nineteen first terms of this progression.
${2}^{19}$
$3.8$
${2}^{10}$
$38$
Question 2:   The second term of the geometric series is $\text{\hspace{0.17em}}{a}_{2}=4$ . Find the product of three first terms of this progression.
$32$
$64$
$16$
$128$
Question 3:   Find the initial term and the common ratio of a geometric series $\text{\hspace{0.17em}}\left({a}_{n}\right)$ , if $\text{\hspace{0.17em}}{a}_{5}=3{a}_{3};\text{\hspace{0.17em}}{a}_{6}-{a}_{2}=48$ .
${a}_{1}=0.2\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=\sqrt{3}$
${a}_{1}=-2\sqrt{3}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=\sqrt{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{1}=2\sqrt{3}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=-\sqrt{3}\text{\hspace{0.17em}}$
${a}_{1}=2\sqrt{3}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=\sqrt{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{1}=-2\sqrt{3}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=-\sqrt{3}$
${a}_{1}=2\sqrt{3}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=\sqrt{3}$
Question 4:   Find the initial term and the common ratio of a geometric sequence which consists of $\text{\hspace{0.17em}}6\text{\hspace{0.17em}}$ terms. The sum of the first three terms is $\text{\hspace{0.17em}}168$ , the sum of the last three terms is $\text{\hspace{0.17em}}21\text{\hspace{0.17em}}$ .
Question 5:   The sum of three positive sequent numbers that make up an arithmetic series $\text{\hspace{0.17em}}\left({a}_{n}\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}21$ . If you add to these numbers, they would make up a geometric series $\text{\hspace{0.17em}}\left({b}_{n}\right)$ . Find these initial numbers.
Question 6:   The sum of three sequent numbers that make up a geometric series $\text{\hspace{0.17em}}\left({b}_{n}\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}65$ . If you subtract $\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$ from the first one and $\text{\hspace{0.17em}}19\text{\hspace{0.17em}}$ from the second one, you get an arithmetic series $\text{\hspace{0.17em}}\left({a}_{n}\right)$ . Find the sequence of these initial numbers.
$45;15;5$
$5;15;45$
$5;15;45\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\text{\hspace{0.17em}}45;15;5$
Question 7:   For any natural $\text{\hspace{0.17em}}n$ , the sum of the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms of a geometric series $\text{\hspace{0.17em}}\left({b}_{n}\right)\text{\hspace{0.17em}}$ is given by a formula $\text{\hspace{0.17em}}{S}_{n}=6\left({\left(-\frac{1}{2}\right)}^{n}-1\right)$ . Find the fourth term of this progression.
$\frac{9}{8}$
$-\frac{45}{8}$
$-1.125$
$-\frac{9}{16}$
Question 8:   Find the sum of squares of six first terms of a geometric series $\text{\hspace{0.17em}}\left({a}_{n}\right)\text{\hspace{0.17em}}$ with the first term $\text{\hspace{0.17em}}{a}_{1}=2\sqrt{3}\text{\hspace{0.17em}}$ and a common ratio $\text{\hspace{0.17em}}r=\sqrt{3}$ .
$4368$
$1452$
$2028×{\left(\sqrt{3}+1\right)}^{2}$
Question 9:   Find the sum of cubes of the first four terms of a geometric series $\text{\hspace{0.17em}}\left({b}_{n}\right)\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}{b}_{1}=3;\text{\hspace{0.17em}}{b}_{2}=-6$ .
$1539$
$-12285$
$15795$
Question 10:   Find the sum of the infinite geometric series $\text{\hspace{0.17em}}\left({a}_{n}\right)\text{\hspace{0.17em}}$ if .
$27-9\sqrt{3}$
$27$