Question #339609

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


1
Expert's answer
2022-05-12T16:56:03-0400

bn=b1qn1b_n=b_1q^{n-1}

b11=b1q10    b1=b11q10b_{11}=b_1q^{10}\iff b_1=\frac{b_{11}}{q^{10}}

b14=b1q13    b14=b11q10q13=b11q3    q3=b14b11b_{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}}

q3=813=127q^3=\frac{81}{3}=\frac{1}{27}

q=1273=13q=\sqrt[3]{\frac{1}{27}}=\frac{1}{3}

b1=811310=34310=314b_1=\frac{81}{\frac{1}{3}^{10}}=3^4*3^{10}=3^{14}

b4=b1q3=31433=311=177147b_4=b_1q^3=3^{14}3^{-3}=3^{11}=177147


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