The ancient Babylonians developed a method for calculating non-perfect squares by 1700 BCE. Complete the statements to demonstrate how to use this method to find the approximate value of [tex]\sqrt{5}[/tex].
1. In order to determine [tex]\sqrt{5}[/tex], let [tex]G_1 = 2[/tex], a number whose square is close to 5.
2. [tex]5 \div G_1 = \square[/tex], which is not equal to [tex]G_1[/tex], so further action is necessary.
3. Average 2 and [tex]5 \div G_1[/tex] to find [tex]G_2 = 2.25[/tex].
4. [tex]5 \div G_2 \approx \square[/tex] (rounded to the nearest thousandth), which is not equal to [tex]G_2[/tex], so further action is necessary.
5. Average 2.25 and [tex]5 \div G_2[/tex] to find [tex]G_3 = 2.236[/tex].
6. [tex]5 \div G_3 \approx \square[/tex] (rounded to the nearest thousandth), which is equal to [tex]G_3[/tex]. That means [tex]\sqrt{5}[/tex] is approximately 2.236.