x, x+5, x+9 form a geometric sequence; find the value(s) of x and the numerical value of each term.
for a GS (x+5)/x = (x+9)/(x+5)
x^2 + 10x + 25 = x^2 + 9x
x = -25
so the terms are -25,-20,-16
Check: -20/-25 = .8
-16/-20 = .8
To determine the value(s) of x and the numerical value of each term in the geometric sequence x, x+5, x+9, we need to establish the relationship between these terms.
In a geometric sequence, each term is obtained by multiplying the preceding term by a constant called the common ratio. Let's assume that the common ratio is r.
So, the terms in the sequence can be represented as:
Term 1: x
Term 2: x * r = x + 5
Term 3: (x + 5) * r = x + 9
Now, we can set up equations to solve for the values of x and r.
First, we equate Term 2 and Term 3:
x * r = x + 5
(x + 5) * r = x + 9
Next, we can solve these equations simultaneously. By substituting the value of x from the first equation into the second equation, we can obtain a single equation in terms of r:
(x + 5) * r = x + 9
(x * r) + (5 * r) = x + 9
(x + 5r) = x + 9
5r = 9
Dividing both sides of the equation by 5, we find:
r = 9/5 = 1.8
Now that we have the value of r, we can substitute it back into the first equation to solve for x:
x * r = x + 5
x * 1.8 = x + 5
1.8x = x + 5
0.8x = 5
Dividing both sides of the equation by 0.8, we get:
x = 5 / 0.8 = 6.25
Thus, the value of x is 6.25 and the numerical value of each term in the geometric sequence is:
Term 1: x = 6.25
Term 2: x + 5 = 6.25 + 5 = 11.25
Term 3: x + 9 = 6.25 + 9 = 15.25
To find the value(s) of x and the numerical value of each term in the geometric sequence x, x+5, x+9, we can use the definition of a geometric sequence.
In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio. Let's denote the common ratio as r.
From the given sequence, we have:
Term 1 = x
Term 2 = x + 5
Term 3 = x + 9
Using the definition, we can set up the following equations:
Term 2 / Term 1 = r
Term 3 / Term 2 = r
Substituting the values from our sequence, we get:
(x + 5) / x = r ............ (Equation 1)
(x + 9) / (x + 5) = r ............ (Equation 2)
To solve this system of equations, we can cross-multiply:
Equation 1: (x + 5) = rx
Equation 2: (x + 9) = r(x + 5)
Expanding the equations gives:
Equation 1: x + 5 = rx
Equation 2: x + 9 = rx + 5r
Now let's rearrange the equations and simplify:
Equation 1: x - rx = -5 ............ (Equation 3)
Equation 2: x - rx = -9 + 5r
Both Equation 1 and Equation 2 give us x - rx, so we can set them equal to each other:
-5 = -9 + 5r
Simplifying further, we obtain:
4 = 5r
Now solve for r:
r = 4 / 5
Next, plug this value of r back into Equation 1:
x - (4 / 5) * x = -5
Multiply through by 5 to eliminate the fraction:
5x - 4x = -25
x = -25
So the value of x is -25.
Now, substitute x = -25 into the original sequence to find the numerical value of each term:
Term 1 = x = -25
Term 2 = x + 5 = -25 + 5 = -20
Term 3 = x + 9 = -25 + 9 = -16
Therefore, the value of x is -25, and the numerical values of the terms in the geometric sequence are -25, -20, -16.