Question

x-4 ; x+2 ; 3x+1 ; ... are the first three terms of a geometric sequence Determine the sequence if x is positive.​

Answer 1

Answer:

• 4, 10, 25, 62.5, ...

Step-by-step explanation:

Given GP:

• x - 4, x + 2, 3x + 1, ...

Common ratio is:

• (x + 2)/(x - 4) = (3x + 1)/(x + 2)

Cross-multiply and solve for positive x:

• (x + 2)² = (x - 4)(3x + 1)
• x² + 4x + 4 = 3x² - 11x - 4
• 2x² - 15x - 8 = 0
• x = (15 + √(15² + 4*2*8))/4
• x = (15 + 17)/4
• x = 8

The first term is:

• 8 - 4 = 4

The common ratio is:

• (8 + 2)/4 = 2.5

The nth term is:

• aₙ = 4*(2.5)ⁿ ⁻ ¹

The sequence is:

• 4, 10, 25, 62.5, ...

Answer 2

Answer:

x = 8

Step-by-step explanation:

This sequence is a geometric sequence.

As it is a geometric sequence, It means that each term multiply by a value that we call that value "a".

So (x-4) × a = (x + 2)

Therefore $\frac{x+2}{x-4} = a$

And (x+2) × a = (3x+1)

Therefore $\frac{3x+1}{x+2} =a$.

By this we can make a conclusion that $\frac{3x+1}{x+2} =\frac{x+2}{x-4}$.

1. $\frac{3x+1}{x+2}= \frac{x+2}{x-4}$
2. $(3x + 1) * (x-4) = (x+2) * (x+2)$
3. $3x^{2} -11x-4=x^{2} +4x+4$
4. $2x^{2} -15x-8 = 0$
5. $(2x+1) * (x-8) = 0$

In the last part one of the parentheses should be 0.

First we put (2x+1) equal to 0.

1. $2x+1 = 0$
2. $2x = -1$
3. $x = -\frac{1}{2}$

It gives us a negative answer, so this is not our answer.

Now we put (x-8) equal to 0.

1. $x-8=0$
2. $x = 8$

This is a positive answer and it is correct.

and the terms are 4, 10, 25 , ...

Please choose my answer as the brainliest answer.