$a)S_3=a_1\cdot \frac{1-q^2}{1-q}$  

$S_6=a_1\cdot \frac{1-q^4}{1-q}$  

 

$S_6=S_3+51,2\%\cdot S_3$ $S_6=S_3+51,2\%\cdot S_3$  

$a_1\frac{1-q^6}{1-q}=a_1\frac{1-q^3}{1-q}+0,512a_1\frac{1-q^3}{1-q}$  

$a_1\frac{1-q^6}{1-q}=1,512a_1\frac{1-q^3}{1-q}$  

$\frac{1-q^6}{1-q^3}=1,512$  

$\frac{\left(1-q^3\right)\left(1+q^3\right)}{1-q^3}=1,512$  

$1+q^3=1,512$  

$q^3=0,512$  

$q^3=\frac{512}{1000}$  

$q^3=\frac{2^9}{{10}^3}$  

$q^3={\left(\frac{2^3}{10}\right)}^3$  

$q=\frac{8}{10}$  

 

$q=0,8$  

 

$b)a_7=25$  

$a_1{q}^6=25$  

$a_1=\frac{25}{{\left(\frac{4}{5}\right)}^6}$  

$a_1=25\cdot {\left(\frac{5}{4}\right)}^6$