$a_1$	$q$	$n$	$a_n$	$S_n$
$4$	$2$	$8$	$512$	$1020$
$27$	$\frac{2}{3}$	$4$	$8$	$65$
$100$	$0,1$	$5$	$0,01$	$111,1$
$\sqrt{3}$	$-\sqrt{3}$	$7$	$27\sqrt{3}$	$40\sqrt{3}-39$

Pierwszy wiersz:

$S_n=a_1\cdot \frac{1-q^8}{1-q}$   

$a_1\cdot \frac{1-256}{1-2}=1020$  

$a_1\cdot 255=1020$  

$a_1=4$  

 

$a_n=a_1\cdot q^{n-1}=4\cdot 2^{n-1}=2^2\cdot 2^{n-1}=2^{n+1}$  

$a_8=2^{8+1}=2^9=512$  

 

Drugi wiersz:

$a_4=8$  

$a_1=27$  

$\frac{a_4}{a_1}=q^3=\frac{8}{27}$  

$q=\frac{2}{3}$  

 

$S_4=a_1\cdot \frac{1-q^4}{1-q}=27\cdot \frac{1-\frac{16}{81}}{\frac{1}{3}}=27\cdot \left(\frac{65}{81}\right)\cdot 3=27\cdot \frac{65}{27}=65$  

 

Trzeci wiersz:

$a_n=a_1\cdot q^{n-1}=100\cdot {\left(0,1\right)}^{n-1}={10}^2\cdot {\left({10}^{-1}\right)}^{n-1}={10}^{2-n+1}={10}^{3-n}$

 

$a_5={10}^{3-5}={10}^{-2}=\frac{1}{100}=0,01$  

 

$S_5=a_1\cdot \frac{1-q^5}{1-q}=100\cdot \frac{1-{\left(0,1\right)}^5}{1-0,1}=100\cdot \frac{10}{9}\cdot \frac{9999}{10000}=\frac{1111}{10}=111,1$  

 

Czwarty wiersz:

$a_n=a_1\cdot q^{n-1}=\sqrt{3}\cdot {\left(-\sqrt{3}\right)}^{n-1}=\sqrt{3}\cdot {\left(-1\right)}^{n-1}\cdot {\left(\sqrt{3}\right)}^{n-1}={\left(-1\right)}^{n-1}\cdot {\left(\sqrt{3}\right)}^n$  

 

$a_7={\left(-1\right)}^{7-1}\cdot {\left(\sqrt{3}\right)}^7={\left(-1\right)}^6\cdot 27\sqrt{3}=27\sqrt{3}$  

 

$S_7=a_1\cdot \frac{1-q^7}{1-q}$  

$S_7=\sqrt{3}\cdot \frac{1+27\sqrt{3}}{1+\sqrt{3}}=\frac{\sqrt{3}+27\cdot 3}{1+\sqrt{3}}\cdot \frac{1-\sqrt{3}}{1-\sqrt{3}}=\frac{\left(\sqrt{3}+81\right)\left(1-\sqrt{3}\right)}{1-3}=\frac{\sqrt{3}-3+81-81\sqrt{3}}{-2}=40\sqrt{3}-39$