Question

Într-o progresie geometrica cu termeni pozitivi se cunosc b1=1 și b3=1/2
b2013=?

Deci după formula b2013=b1*q^n-1,adică b2013=1*q^2012.
Nu înțeleg cum aflu q? ​

Answer 1

 

$\displaystyle\\b_2=b_1\times q\\\\b_3=b_2\times q=b_1\times q\times q\\\\b_3=b_1\times q^2\\\\q^2=\frac{b_3}{b_1}\\\\\\q^2=\frac{~\dfrac{1}{2}~}{1}\\\\q^2=\frac{1}{2}\\\\q=\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}}\\\\b_{2013}=b_1\times q^{2013-1}\\\\b_{2013}=b_1\times q^{2012}\\\\b_{2013}=1\times \left(\frac{1}{\sqrt{2}}\right)^{2012}\\\\\\b_{2013}=1\times \left(\frac{1}{\sqrt{2}}\right)^{2\times1006}\\\\\\b_{2013}=1\times\left[\left(\frac{1}{\sqrt{2}}\right)^2\right]^{1006}$

.

$\displaystyle\\b_{2013}=1\times\left(\frac{1}{2}\right)^{1006}\\\\\\b_{2013}=1\times\frac{1}{2^{1006}}\\\\\\\boxed{\bf~b_{2013}=\frac{1}{2^{1006}}}$