I NEED URGENT HELP WITH THIS QUESTION!!!!
Question:
The terms of a geometric sequence are 4x+1, x+4, and 10-x. Determine the value of x.
since there is a common ratio between terms,
(x+4)/(4x+1) = (10-x)/(x+4)
Now just solve for x.
Use your definition.
(x+4)/(4x+1) = (10-x)/(x+4)
(x+4)^2 = (4x+1)(10-x)
x^2 + 8x + 16 = 40x - 4x^2 + 10 - x
5x^2 - 31x + 6 = 0
solve for x, it factors
Did that and it didn’t work. I am not getting the correct value for x. x = 6.
It would be great if someone showed me the steps on how to solve the question
To determine the value of x in the given geometric sequence, we need to use the properties of geometric sequences. A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Let's use this information to solve the question step by step:
Step 1: Determine the common ratio (r):
In a geometric sequence, the common ratio (r) is found by dividing any term (except the first) by its preceding term. Let's use the second and first terms to find the common ratio:
(r) = (x+4)/(4x+1)
Step 2: Equate the common ratio to the ratio of the third and second terms:
Since it is a geometric sequence, the common ratio (r) should be the same when we divide the third term by the second term. Let's set up the equation:
(x+4)/(4x+1) = (10-x)/(x+4)
Step 3: Solve the equation:
To solve the equation, we can first cross-multiply:
(x+4)(x+4) = (10-x)(4x+1)
Expand both sides of the equation:
x^2 + 8x + 16 = 40x + 10 - 4x^2 - x
Combine like terms:
5x^2 + 33x - 26 = 0
Step 4: Solve the quadratic equation:
To find the value of x, we can solve the quadratic equation. You can either factor the quadratic or use the quadratic formula. However, since this equation might not factor easily, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 5, b = 33, and c = -26.
Plugging in the values and simplifying, we get:
x = (-33 ± √(33^2 - 4 * 5 * -26)) / (2 * 5)
x = (-33 ± √(1089 + 520)) / 10
x = (-33 ± √(1609)) / 10
Step 5: Solve for x:
Now, we have two solutions for x:
x = (-33 + √1609) / 10
x = (-33 - √1609) / 10
These are the values of x that satisfy the given geometric sequence.
I told you it factored
5x^2 - 31x + 6 = 0
(x-6)(5x-1) = 0
x = 6 or x = 1/5
if x = 6, your terms are: 25, 10, and 4
notice 10/25 = 4/10 = 2/5
if x = 1/5, your terms are : 9/5, 21/5, and 49/5
is (21/5) ÷ (9/5) = (49/5) ÷ (21/5) ??