a) Mamy dane

$a_6\cdot a_8\cdot a_{10}\cdot a_{12}=81$  

Rozpiszmy ze wzoru każdy wyraz

$a_1{q}^5\cdot a_1{q}^7\cdot a_1{q}^9\cdot a_1{q}^{11}=81$  

$a_1^4{q}^{32}=81$  

${\left(a_1{q}^8\right)}^4=3^4$  

$a_1{q}^8=3$  

Stąd

$a_9=3$  

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b) Mamy dane

$a_9\cdot a_{13}\cdot a_{17}\cdot a_{21}=10000$  

Rozpiszmy ze wzoru każdy wyraz

$a_1{q}^8\cdot a_1{q}^{12}\cdot a_1{q}^{16}\cdot a_1{q}^{20}=10000$  

$a_1^4{q}^{56}=10000$  

${\left(a_1{q}^{14}\right)}^4={10}^4$  

$a_1{q}^{14}=10{\mid }^{\left(5\right)}$  

${\left(a_1{q}^{14}\right)}^5={10}^5$  

$a_1^5{q}^{70}=100000$  

$a_1{q}^4\cdot a_1{q}^9\cdot a_1{q}^{14}\cdot a_1{q}^{19}\cdot a_1{q}^{24}=100000$

$a_5\cdot a_{10}\cdot a_{15}\cdot a_{20}\cdot a_{25}=100000$

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c) Mamy dane

$a_2\cdot a_9\cdot a_{16}=2\sqrt{2}$  

Rozpiszmy ze wzoru

$a_1{q}^1\cdot a_1{q}^8\cdot a_1{q}^{15}=2\sqrt{2}$  

$a_1^3{q}^{24}=2\sqrt{2}$  

${\left(a_1{q}^8\right)}^3={\left(\sqrt{2}\right)}^3$  

$a_1{q}^8=\sqrt{2}{\mid }^{\left(5\right)}$  

${\left(a_1{q}^8\right)}^5={\left(\sqrt{2}\right)}^5$  

$a_1^5{q}^{40}=4\sqrt{2}$  

$a_1{q}^2\cdot a_1{q}^5\cdot a_1{q}^8\cdot a_1{q}^{11}\cdot a_1{q}^{14}=4\sqrt{2}$  

$a_3\cdot a_6\cdot a_9\cdot a_{12}\cdot a_{15}=4\sqrt{2}$