Answer:
The geometric mean of the sequence is 1814.99.
Step-by-step explanation:
The formula to compute the geometric mean is:
[tex]GM=\sqrt[n]{a_{1}\cdot a_{2}\cdot a_{3}\cdot...a_{n}}[/tex]
The geometric sequence is:
–14, ? , ? , ? , ? , –235,298
The nth term of a geometric sequence is:
[tex]a_{n}=a_{1}\cdot r^{n-1}[/tex]
Here, r is the common ratio.
There are 6 terms in the sequence provided.
Compute the common ratio as follows:
[tex]a_{6}=a_{1}\cdot r^{6-1}\\\\-235298=-14\times r^{5}\\\\r=[\frac{235298}{14}]^{1/5}\\\\r=7[/tex]
Thus, the common ratio is 7.
The missing terms are:
[tex]a_{2}=a_{1}\cdot r^{2-1}\\=-14\times 7\\=-98[/tex]
[tex]a_{3}=a_{1}\cdot r^{3-1}\\=-14\times 7^{2}\\=-686[/tex]
[tex]a_{4}=a_{1}\cdot r^{4-1}\\=-14\times 7^{3}\\=-4802[/tex]
[tex]a_{5}=a_{1}\cdot r^{5-1}\\=-14\times 7^{4}\\=-33,614[/tex]
Compute the geometric mean as follows:
[tex]GM=\sqrt[n]{a_{1}\cdot a_{2}\cdot a_{3}\cdot...a_{n}}[/tex]
[tex]=\sqrt[6]{-14\times -98\times -686\times -4802\times-33614\times -235298} \\\\=1814.9854\\\\\approx 1814.99[/tex]
Thus, the geometric mean of the sequence is 1814.99.
