# Test: Geometric Series I - Ambitious

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Question 1:   Express the term of a geometric progression $\text{\hspace{0.17em}}{a}_{18}\text{\hspace{0.17em}}$ through its another term $\text{\hspace{0.17em}}{a}_{12}\text{\hspace{0.17em}}$ and its common ratio $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ .
${a}_{18}={a}_{12}×{r}^{6}$
${a}_{18}={a}_{12}×6r$
${a}_{18}=\frac{{a}_{12}}{{r}^{6}}$
${a}_{18}={a}_{12}+6r$
Question 2:   ($\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ ) is a geometric sequence. Put the sign between these two expressions: .
Question 3:   Find the initial term of the geometric sequence ($\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ ), if $\text{\hspace{0.17em}}r=\frac{1}{2}\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}{a}_{4}=\frac{1}{32}\text{\hspace{0.17em}}$ .
$4$
$\frac{1}{4}$
$\frac{1}{8}$
$\frac{1}{2}$
Question 4:   $192\text{\hspace{0.17em}}$ is a term of a geometric sequence: . Find its number.
$10$
$6$
$7$
$5$
Question 5:   What two numbers should be put between $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}192\text{\hspace{0.17em}}$ two create a geometric progression?
$66;\text{\hspace{0.17em}}129$
$-12;\text{\hspace{0.17em}}48$
$12;\text{\hspace{0.17em}}48$
$52;\text{\hspace{0.17em}}103$
Question 6:   Find the sum of the first four terms of the geometric sequence ($\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ ) if .
$70$
$3780$
$\frac{560}{3}$
$140$
Question 7:   Find the sum of the first six terms of the geometric sequence ($\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ ), if $\text{\hspace{0.17em}}{a}_{1}=5,\text{\hspace{0.17em}}{a}_{5}=125,\text{\hspace{0.17em}}r>0\text{\hspace{0.17em}}$ .
$155\left(\sqrt{5}+1\right)$
$155\left(\sqrt{5}+5\right)$
$\frac{\sqrt{5}-1}{620}$
$155$
Question 8:   Geometric sequence is given by a formula $\text{\hspace{0.17em}}{a}_{n}=10×{3}^{n}\text{\hspace{0.17em}}$ . Find the sum of the first five terms.
$2400$
$1200$
$2430$
$900$
Question 9:   The common ratio of the geometric progression ($\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ ) is $\text{\hspace{0.17em}}r=2\text{\hspace{0.17em}}$ . Find the initial term if the sum of the first four terms equals $\text{\hspace{0.17em}}60\text{\hspace{0.17em}}$ .
$2$
$8$
$1$
$4$
Question 10:   Find the sum of the infinite geometric progression ($\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ ), if .
$72$
$27$
$81$
$162$