Question

4. Use the process outlined in the lesson to approximate the number 2√3. Use the approximation √3 ≈ 1.732 050 8.

b. Find a sequence of five intervals that contain 2√3 whose endpoints get successively closer to 2√3. Write your
intervals in the form 2???????? < 2√3 < 2ss for rational numbers ???????? and ss.

Answer 1

sequence of five intervals

(1) 2 <
`2^(√(3) )` <
`2^(2)`

(2)
`2^(1.7)` <
`2^(√(3) )` <
`2^(1.8)`

(3)
`2^(1.73)` <
`2^(√(3) )` <
`2^(1.74)`

(4)
`2^(1.732)` <
`2^(√(3) )` <
`2^(1.733)`

(5)
`2^(1.7320)` <
`2^(√(3) )` <
`2^(1.7321)`

Explanation:

as per question given data

√3 ≈ 1.732 050 8

to find out

sequence of five intervals

solution

as we have given that √3 value that is here

√3 ≈ 1.732 050 8 ........................1

so

when we find
`2^(√(3) )` ................2

put here √3 value in equation number 2

we get
`2^(√(3) )` that is 3.322

so

sequence of five intervals

(1) 2 <
`2^(√(3) )` <
`2^(2)`

(2)
`2^(1.7)` <
`2^(√(3) )` <
`2^(1.8)`

(3)
`2^(1.73)` <
`2^(√(3) )` <
`2^(1.74)`

(4)
`2^(1.732)` <
`2^(√(3) )` <
`2^(1.733)`

(5)
`2^(1.7320)` <
`2^(√(3) )` <
`2^(1.7321)`