find the three geometric means between -2/3 and -54
hint: r = 3
meko nahi ata b.c
To find the three geometric means between two numbers, we need to find a common ratio between successive terms and then apply the formula for geometric mean.
First, let's find the common ratio between -2/3 and -54. The common ratio, denoted by "r", is found by dividing the second term by the first term:
r = (-54) / (-2/3)
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
r = (-54) * (3 / -2)
r = 162 / -2
r = -81
Now, we can use the formula for geometric mean (denoted by "G") to find the three geometric means:
G1 = √((-2/3) * (-81))
G2 = √((-2/3) * (-81) * (-81))
G3 = √((-2/3) * (-81) * (-81) * (-81))
Let's calculate these values:
G1 = √((-2/3) * (-81))
= √(162/3)
= √54
= 2√3
G2 = √((-2/3) * (-81) * (-81))
= √(6561/3)
= √2187
= 3√3
G3 = √((-2/3) * (-81) * (-81) * (-81))
= √(531441/3)
= √177147
= 6√3
Therefore, the three geometric means between -2/3 and -54 are 2√3, 3√3, and 6√3.