The sum of the first two terms of a G.P is 2 and the half and the sum of the first four terms is 3 11/18 find the G.P if r =is less than. 0
Let the first term of the geometric progression be a, and the common ratio be r.
The sum of the first two terms is a + ar = 2.
The sum of the first four terms is a + ar + ar^2 + ar^3 = 3 11/18.
We can rewrite the second equation as:
(1 + r + r^2 + r^3)a = 3 11/18.
Dividing both sides by 2:
(1/2)(1 + r + r^2 + r^3)a = 3 11/36.
Since r < 0, we can multiply both sides by -1 without changing the inequality:
(-1/2)(1 + r + r^2 + r^3)a = -3 11/36.
(-1/2)(1 + r + r^2 + r^3) = -3 11/36a.
(1 + r + r^2 + r^3) = 6 22/36.
(1 + r + r^2 + r^3) = 6 11/18.
r^3 + r^2 + r + 1 = 6 11/18.
r^3 + r^2 + r + 1 = 6 22/36.
r^3 + r^2 + r + 1 = 7 11/18.
We can rewrite 7 11/18 as 131/18.
r^3 + r^2 + r + 1 = 131/18.
18r^3 + 18r^2 + 18r + 18 = 131.
18r^3 + 18r^2 + 18r - 113 = 0.
We can use trial and error to find one of the roots of this polynomial. Let's try r = -2:
18(-2)^3 + 18(-2)^2 + 18(-2) - 113 = 0.
-144 + 72 - 36 - 113 = 0.
-144 - 36 - 141 = 0.
-321 = 0.
This is not true, so r = -2 is not a root.
We can try another value, r = -3:
18(-3)^3 + 18(-3)^2 + 18(-3) - 113 = 0.
-486 + 162 - 54 - 113 = 0.
-571 = 0.
This is not true, so r = -3 is not a root.
We can try another value, r = -4:
18(-4)^3 + 18(-4)^2 + 18(-4) - 113 = 0.
-1152 + 288 - 72 - 113 = 0.
-1049 = 0.
This is not true, so r = -4 is not a root.
We can try another value, r = -5:
18(-5)^3 + 18(-5)^2 + 18(-5) - 113 = 0.
-2250 + 450 - 90 - 113 = 0.
-2003 = 0.
This is not true, so r = -5 is not a root.
We can try another value, r = -6:
18(-6)^3 + 18(-6)^2 + 18(-6) - 113 = 0.
-1944 + 648 - 108 - 113 = 0.
-1517 = 0.
This is not true, so r = -6 is not a root.
We can try another value, r = -7:
18(-7)^3 + 18(-7)^2 + 18(-7) - 113 = 0.
-2646 + 882 - 126 - 113 = 0.
-2003 = 0.
This is not true, so r = -7 is not a root.
Based on the values we've tried, we can see that there are no roots for this polynomial when r < 0.
Thus, there is no solution for the G.P with r < 0.