# 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.
$38$
${2}^{10}$
${2}^{19}$
$3.8$
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.
$128$
$32$
$16$
$64$
Question 3:   Find the initial term and the common ratio of a geometric series $\text{\hspace{0.17em}}\left({a}_{n}\right)$ , if $\text{\hspace{0.17em}}{a}_{5}=3{a}_{3};\text{\hspace{0.17em}}{a}_{6}-{a}_{2}=48$ .
${a}_{1}=2\sqrt{3}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=\sqrt{3}$
${a}_{1}=-2\sqrt{3}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=\sqrt{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{1}=-2\sqrt{3}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=-\sqrt{3}$
${a}_{1}=0.2\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}r=\sqrt{3}$
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.
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