Find the value of x for which x+9 x-6 4 are the first three terms of a geometrical progression and calculate the fourth term of progression in each case
they have a common ratio, so
(x-6)/(x+9) = 4/(x-6)
You will find two values of x which work.
X=9
To find the value of x for which x+9, x-6, and 4 are the first three terms of a geometric progression, we can use the formula for a geometric progression.
The formula for the nth term of a geometric progression is given by:
\(a_n = a_1 \times r^{(n-1)}\)
Where:
- \(a_n\) is the nth term
- \(a_1\) is the first term
- r is the common ratio
- n is the position of the term
In this case, we are given the first three terms: \(a_1 = x+9\) (first term), \(a_2 = x-6\) (second term), and \(a_3 = 4\) (third term). We need to find the value of x and calculate the fourth term.
Using the formula, we can write the following equations:
\(a_2 = a_1 \times r^{(2-1)}\) and \(a_3 = a_1 \times r^{(3-1)}\)
Substituting the values, we get:
\(x-6 = (x+9) \times r\) and \(4 = (x+9) \times r^2\)
To solve this system of equations, we can use the substitution or elimination method. For simplicity, let's use the substitution method:
From the first equation, we can isolate x by bringing the -6 to the other side:
\(x = 6 + (x+9) \times r\)
Now, substitute this value of x into the second equation:
\(4 = (6 + (x+9) \times r + 9) \times r^2\)
Expand and simplify the equation:
\(4 = (15r + 15) \times r^2\)
Now, let's solve for r by factoring out 15r from the right side of the equation:
\(4 = 15r \times (r + 1)\)
Divide both sides by 15r:
\(\frac{4}{15r} = r + 1\)
Subtract 1 from both sides:
\(\frac{4}{15r} - 1 = r\)
Now, we can substitute this value of r back into the first equation to solve for x:
\(x = 6 + (x + 9) \times \left(\frac{4}{15r} - 1\right)\)
Simplify and solve for x:
\(x = 6 + \frac{4(x+9)}{15} - x - 9\)
Combine like terms:
\(x = \frac{4x + 36}{15} - x - 3\)
Multiply both sides by 15 to eliminate the fraction:
\(15x = 4x + 36 - 15x - 45\)
Simplify:
\(15x - 4x = 36 - 45\)
Combine like terms:
\(11x = -9\)
Divide both sides by 11:
\(x = -\frac{9}{11}\)
So, the value of x for which x+9, x-6, and 4 are the first three terms of a geometric progression is -9/11.
To calculate the fourth term of the progression, we can substitute this value of x into the formula for the nth term:
\(a_4 = (x + 9) \times r^3\)
Substituting x = -9/11 and the value of r we found earlier, we get:
\(a_4 = \left(-\frac{9}{11} + 9\right) \times \left(\frac{4}{15 \times \left(-\frac{9}{11}\right)} - 1\right)^3\)
Simplifying this expression will give you the fourth term of the progression.