Answer to Question #339609 in Algebra for Lamek

Question

the 11th term of a geometric sequence is 81 and the 14th term is 3. find the 4th term

Answer 1

$b_n=b_1q^{n-1}$

$b_{11}=b_1q^{10}\iff b_1=\frac{b_{11}}{q^{10}}$

$b_{14}=b_1q^{13}\iff b_{14}=\frac{b_{11}}{q^{10}}*q^{13}=b_{11}q^3\iff q^3=\frac{b_{14}}{b_{11}}$

$q^3=\frac{81}{3}=\frac{1}{27}$

$q=\sqrt[3]{\frac{1}{27}}=\frac{1}{3}$

$b_1=\frac{81}{\frac{1}{3}^{10}}=3^4*3^{10}=3^{14}$

$b_4=b_1q^3=3^{14}3^{-3}=3^{11}=177147$