Find three geometric means betwee -sqrt(2) and -4sqrt(2).

A: 2, -2sqrt(2), 4

Is 6 sqrt(2)+ 6 + 3sqrt(2)+... a convergent series?

A: Yes.

Is 1 + 3(1/2) + 9(1/2)^2 + 27(1/2)^3 +... a divergent series?

A: Yes.

Are these right?

All correct!

To find three geometric means between two numbers, we need to find the common ratio between those numbers.

In this case, we have -√2 and -4√2. To find the common ratio, we can divide the second number by the first number:

(-4√2) / (-√2) = 4

Now we can calculate the three geometric means by repeatedly multiplying the common ratio:

-√2 * 4 = -4√2 (first geometric mean)
-4√2 * 4 = -16√2 (second geometric mean)
-16√2 * 4 = -64√2 (third geometric mean)

So, the three geometric means between -√2 and -4√2 are -4√2, -16√2, and -64√2.

Regarding the second question about the series 6√2 + 6 + 3√2 + ..., we need to determine if this series converges or diverges.

To analyze the convergence of the series, we can check the behavior of the common ratio. In this case, the common ratio between the terms is not constant. It alternates between √2 and 1.

Since the common ratio is not constant, the series does not converge. Therefore, the statement is incorrect, and the answer is no, the series is not convergent.

Regarding the third question about the series 1 + 3(1/2) + 9(1/2)^2 + 27(1/2)^3 + ..., this is a geometric series with a common ratio of (1/2).

To determine if the series converges or diverges, we examine the absolute value of the common ratio, which is (1/2).

If the absolute value of the common ratio is less than 1, the series converges. In this case, the absolute value of (1/2) is less than 1, so the series converges.

Therefore, the statement is incorrect, and the answer is yes, the series 1 + 3(1/2) + 9(1/2)^2 + 27(1/2)^3 + ... is a convergent series.