Answer: 21, 63
Step-by-step explanation:
We need to find two geometric means between 7 and 189.
The Geometric Sequence is: 7, ____, ____, 189
Let's find r (common ratio):
We can see that the common ratio will be multiplied by itsely 3 times in order to get from 7 to 189:
7 r³ = 189
[tex]r^3=\dfrac{189}{7}[/tex]
r³ = 27
∛r³ = ∛27
r = 3
Now that we know the common ratio is 3, we can multiply 7 and 3 to get the next term (21), and multiply that by 3 to get the third term (63), and multiply that by 3 as a check to confirm we get 189.
7 x 3 = 21
21 x 3 = 63
63 x 3 = 189 [tex]\checkmark[/tex]
The completed Geometric Sequence is: 7, 21, 63, 189
We have that the two geometric means between 7 and 189 are
[tex]x=21 \ and \ y=63[/tex]
From the calculations below
From the question we are told that:
There are two geometric means
7 and 189
Generally for two geometric means between 7 and 189
Let the unknown means be x and y respectively
Therefore
The geometric Progression will be
7,x,y,189
Since they share a similar common ratio r
Therefore
[tex]r=\frac{x}{7}=\frac{y}{x}=\frac{189}{y}[/tex]
Hence
[tex]\frac{x}{7}=\frac{y}{x}\\\\y=\frac{x^2}{7}..........equ1[/tex]
And
[tex]\frac{y}{x}=\frac{189}{y}\\\\y^2=189x.........equ 2[/tex]
Therefore
Substituting equ1 into equ2 we have
[tex](\frac{x^2}{7})^2=189x\\\\x=21[/tex]
Having x we substitute x into equ...1
[tex]y=\frac{(21)^2}{7}\\\\y=63[/tex]
In Conclusion
The two geometric means between 7 and 189
[tex]x=21 \ and \ y=63[/tex]
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In conclusion
