Answer:
we can conclude that the terms √3 - √2, 4 - √6, and 6√3 - 2√2 do not form a geometric sequence.
Step-by-step explanation:
To determine whether the given terms form a geometric sequence, we need to check if the ratio between consecutive terms is constant.
Here's how we can solve the problem step-by-step:
1. Let's write down the three terms given:
Term 1: √3 - √2
Term 2: 4 - √6
Term 3: 6√3 - 2√2
2. To find the ratio between consecutive terms, we divide each term by the previous one.
Ratio between Term 2 and Term 1: (4 - √6) / (√3 - √2)
Ratio between Term 3 and Term 2: (6√3 - 2√2) / (4 - √6)
3. Simplify each ratio:
Ratio between Term 2 and Term 1:
(4 - √6) / (√3 - √2) = ((4 - √6) / (√3 - √2)) * ((√3 + √2) / (√3 + √2))
= (4√3 + 4√2 - √18 - √12) / (3 - 2)
= (4√3 + 4√2 - √9√2 - √4√3) / 1
= (4√3 + 4√2 - 3√2 - 2√3)
Ratio between Term 3 and Term 2:
(6√3 - 2√2) / (4 - √6) = ((6√3 - 2√2) / (4 - √6)) * ((4 + √6) / (4 + √6))
= (6√3*4 + 6√3√6 - 2√2*4 - 2√2√6) / (16 - 6)
= (24√3 + 6√18 - 8√2 - 2√12) / 10
= (24√3 + 6√9√2 - 8√2 - 2√4√3) / 10
= (24√3 + 6*3√2 - 8√2 - 2*2√3) / 10
= (24√3 + 18√2 - 8√2 - 4√3) / 10
= (20√3 + 10√2) / 10
= 2√3 + √2
4. If the terms form a geometric sequence, the ratio between consecutive terms should be the same. Let's compare the two ratios:
Ratio between Term 2 and Term 1: 4√3 + 4√2 - 3√2 - 2√3
Ratio between Term 3 and Term 2: 2√3 + √2
The two ratios are not the same, which means the given terms do not form a geometric sequence.
Therefore, we can conclude that the terms √3 - √2, 4 - √6, and 6√3 - 2√2 do not form a geometric sequence.
