Question

find three geometric means between - sqrt 2 and -4sqrt 2

Answer 1

a. 2, -2sqrt2, 4
b. -2, 2sqrt2,-4
c. 2, 2sqrt2, 4
d. A or C

Answer 2

"The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc."
https://www.mathsisfun.com/numbers/geometric-mean.html

Answer 3

let g1,g2,g3 be G.M's between - sqrt2 and -4 sqrt2
Then \[-\sqrt{2},g1,g2,g3,-4\sqrt{2}are \in G.P.\]
let r be common ratio.
no.of terms=5
\[a1=-\sqrt{2}\]
\[a5=-4\sqrt{2}\]

Answer 4

\[an=ar ^{n-1}\]
\[a5=a1r ^{5-1}=a1r^4\]
\[-4 \sqrt2=- \sqrt2 r^4,r^4=\frac{ -4 \sqrt2 }{ -\sqrt2 }=4,(r^2)^2=2^2,r^2=2,r=\sqrt2\]

Answer 5

\[g1=a1r=-\sqrt2*\sqrt2=-2\]
\[g2=g1r=-2*\sqrt2=-2\sqrt2\]
\[g3=g2r=-2\sqrt2*\sqrt2=-4\]

Answer 6

r also can be \[-\sqrt2\]
g1=a1r\[=-\sqrt2*-\sqrt2=2\]
\[g2=a1r=2*-\sqrt2=-2\sqrt2\]
\[g3=g2r=-2\sqrt2*-\sqrt2=4\]