The 3 rd,4 th ,5th term of a g.p are x+3,x+8,x+18
Any answer and working pls
since the terms are consecutive ...
(x+8)/(x+3) = (x+18)/(x+8)
(x+8)^2 = (x+3)(x+18)
expand and simplify
the x^2 s will drop out, so will have a unique answer for x
To find the common ratio, we can use the formula for the nth term of a geometric progression (g.p):
\(a_n = a \cdot r^{(n-1)}\)
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
Given that the 3rd, 4th, and 5th terms are \(x+3\), \(x+8\), and \(x+18\) respectively, we can plug in these values into the formula:
For the 3rd term:
\(x + 3 = a \cdot r^{(3-1)}\)
For the 4th term:
\(x + 8 = a \cdot r^{(4-1)}\)
For the 5th term:
\(x + 18 = a \cdot r^{(5-1)}\)
To find the value of \(x\), we can solve the system of equations formed by these three equations. Substituting the first equation into the other two equations, we get:
\(x + 8 = (x + 3) \cdot r\)
and
\(x + 18 = (x + 3) \cdot r^2\)
Expanding and simplifying these equations, we have:
\(x + 8 = xr + 3r\) [equation 1]
and
\(x + 18 = xr^2 + 3r^2\) [equation 2]
Now, we can solve this system of equations step-by-step:
Step 1:
Rearrange equation 1 and substitute it into equation 2:
\(x = 5r - 8\)
Substituting this value of \(x\) into equation 2:
\(5r - 8 + 18 = (5r - 8)r^2 + 3r^2\)
Step 2:
Combine like terms and rewrite the equation in standard form:
\(5r - 8 + 18 = 5r^3 - 8r^2 + 3r^2\)
Simplifying further:
\(5r + 10 = 5r^3 - 5r^2\)
Step 3:
Rearrange the terms to form a cubic equation:
\(5r^3 - 5r^2 - 5r - 10 = 0\)
Step 4:
Factor out a common factor of 5:
\(5(r^3 - r^2 - r - 2) = 0\)
Step 5:
Now, we have a cubic equation. To find the common ratio (\(r\)), we can either solve this equation using numerical methods (such as Newton's method) or factor it further if possible. However, let's assume that the factorization is not readily apparent.
Step 6:
Since we are given that this is a geometric progression, there must be a common ratio that exists. Therefore, one solution for this equation is \(r = 0\).
Step 7:
Now, we can consider the remaining polynomial:
\(r^3 - r^2 - r - 2 = 0\)
To find the remaining solutions, we can use numerical methods or a graphing calculator.
Therefore, we have found one common ratio (\(r = 0\)) and other possible values of \(r\) can be obtained by solving the cubic equation.
To find the common ratio (r) and the first term (a) of the geometric progression (g.p.), we can use the given information.
Let's consider the 3rd and 4th terms of the g.p.
The 3rd term (T3) is x + 3, while the 4th term (T4) is x + 8.
We know that the nth term of a g.p. is given by the formula: Tn = a * r^(n-1), where Tn is the nth term, a is the first term, and r is the common ratio.
Plugging in the values for the 3rd term and the 4th term, we get:
T3 = a * r^(3-1) = x + 3
T4 = a * r^(4-1) = x + 8
Now we can divide the equations to eliminate the first term:
(T4 / T3) = [(a * r^3) / (a * r^2)]
(x + 8) / (x + 3) = r
Similarly, we can use the 4th and 5th terms (T4 and T5) to find another equation involving a and r:
T4 = a * r^(4-1) = x + 8
T5 = a * r^(5-1) = x + 18
Dividing these equations, we get:
(T5 / T4) = [(a * r^4) / (a * r^3)]
(x + 18) / (x + 8) = r
Now we have two equations with r, and we can solve them simultaneously to find the values of r and x.
(x + 8) / (x + 3) = (x + 18) / (x + 8)
We can cross-multiply to simplify the equation:
(x + 8)(x + 8) = (x + 3)(x + 18)
Expanding both sides:
(x^2 + 16x + 64) = (x^2 + 21x + 54)
Now, let's rearrange the equation:
x^2 + 16x + 64 - x^2 - 21x - 54 = 0
Simplifying further:
-5x - 10 = 0
Dividing by -5:
x = 2
Now we can substitute the value of x back into one of the equations to find the common ratio:
(x + 8) / (x + 3) = r
(2 + 8) / (2 + 3) = r
10/5 = r
r = 2
Therefore, the common ratio (r) of the geometric progression is 2, and the first term (a) can be calculated using any of the given terms, such as the 3rd term:
T3 = a * r^(3-1) = x + 3
T3 = a * 2^2 = x + 3
T3 = a * 4 = x + 3
So, a = (x + 3) / 4 = (2 + 3) / 4 = 5/4