The third, the fourth and the fifth terms of a G.P. are (w+6), w,and (w-3) respectively.
a. Find the value of w.
b. Find the common ratio
By definition of a GP, the ratio of successive terms are equal.
So for (w+6), w, and (w-3), we have
(w-3)/w = w/(w+6)
cross multiply to get
w²+3w-18 = w²
Solve for w.
The ratio
(w-3)/w or w/(w+6) is the common ratio.
To find the value of w and the common ratio in this geometric progression (G.P.), we can use the formula for the terms of a G.P.
The general formula for the terms of a G.P. is given by:
An = A1 * r^(n-1),
where An is the nth term, A1 is the first term, r is the common ratio, and n is the term number.
Given that the third term is (w+6), the fourth term is w, and the fifth term is (w-3), we can substitute these values into the formula to create three equations.
For the third term:
(w+6) = A1 * r^(3-1)
(w+6) = A1 * r^2 -----------(1)
For the fourth term:
w = A1 * r^(4-1)
w = A1 * r^3 -----------(2)
For the fifth term:
(w-3) = A1 * r^(5-1)
(w-3) = A1 * r^4 -----------(3)
We can now solve these three equations simultaneously to determine the value of w and the common ratio.
First, let's take equation (1) and equation (2) to eliminate A1:
(w+6) / w = r^2 / r^3
(w+6) / w = 1 / r
Cross-multiplying, we get:
(w+6) * r = w ------------(4)
Next, let's take equation (2) and equation (3) to eliminate A1:
w / (w-3) = r^3 / r^4
w / (w-3) = 1 / r
Cross-multiplying, we get:
w * r = w-3 ---------------(5)
Now, we have two equations (4) and (5) with two variables (w and r). We can solve these equations simultaneously.
Multiplying equation (4) by w, we get:
(w+6) * r * w = w^2
Expanding and rearranging, we have:
w^2 - w*r - 6*r*w = 0
Factoring out w, we get:
w * (w - r - 6r) = 0
Since w ≠ 0, we can divide both sides by (w - r - 6r) to solve for w:
w = 0 or w = r + 6r
As w cannot be 0 (as per the equations given), we have:
w = r + 6r
Simplifying, we get:
w = 7r
Now, substitute this value of w into equation (4):
(w+6) * r = w
(7r + 6) * r = 7r
Expanding, we get:
7r^2 + 6r = 7r
Bringing all terms to one side, we have:
7r^2 - r = 0
Factoring out r, we get:
r * (7r - 1) = 0
Therefore, r = 0 or r = 1/7.
To summarize:
a. The value of w is 7r (since we found w = r + 6r).
b. The common ratio can be either 0 or 1/7.
(Note: The common ratio cannot be zero, as it will lead to all terms being the same (w), and the third, fourth, and fifth terms given in the question would be equal then.)