Find the sixth term in the geometric sequence: sqrt3y^3, -3y^5, 3sqrt3y^7
Answer: -27y^13
2) Find three geometric means between -sqrt2 and -4sqrt2.
Answer: -2, 2sqrt2, -4
Thanks
1) I agree
2) Your answer is OK, but these are not geometric means. They are three numbers in a geometric progression. Another possible answer would be -2, -2 sqrt2, -4.
To find the sixth term in a geometric sequence, we need to find the common ratio and apply it to the first term.
Given the geometric sequence: √3y^3, -3y^5, 3√3y^7, ...
Step 1: Find the common ratio (r)
The common ratio (r) is calculated by dividing any term by its previous term. Let's choose the second term (-3y^5) and the first term (√3y^3):
r = (-3y^5) / (√3y^3)
Step 2: Simplify the expression
To simplify this expression, we'll rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator (√3y^3):
r = [(-3y^5) * (√3y^3)] / [(√3y^3) * (√3y^3)]
r = (-3y^5 * √3y^3) / (3y^6)
Simplifying further:
r = -√3 / y
Step 3: Apply the common ratio
To find the sixth term, we need to apply the common ratio (r) five times (since we know the first three terms). Starting with the third term (3√3y^7), we apply the common ratio four more times:
-3y^5 * (-√3 / y)^4
Step 4: Simplify the expression
Using the exponent rule for dividing powers,
-3y^5 * (-√3 / y)^4 = -3y^5 * (√3)^4 / y^4
Simplifying further,
=(-3 • 3^2 • y^5 • y^4) / y^4
=(-3 • 9 • y^5) / 1
=-27y^5
Therefore, the sixth term in the geometric sequence is -27y^5.
Regarding the second question, to find three geometric means between two given terms, we need to find the common ratio (r) by taking the n-th root of the ratio of the second term to the first term, where n is the number of geometric means plus 1.
Given: -√2, -4√2
Step 1: Find the common ratio (r)
r = (√2) / (-√2)
r = 2 / -2
r = -1
Step 2: Find the three geometric means
To find three geometric means, we apply the common ratio (-1) successively.
First geometric mean = (-√2) * (-1) = √2
Second geometric mean = √2 * (-1) = -√2
Third geometric mean = -√2 * (-1) = √2
Therefore, the three geometric means between -√2 and -4√2 are: -2, 2√2, -4.
If you have any further questions or need more clarification, please let me know!