The sum of the first two terms of a G.P. is 25, and the sum of the first four terms is
11
3
18
Find the G.P. if r > 0.
Let the first term of the G.P. be a and the common ratio be r.
The sum of the first two terms is 25, so we have:
a + ar = 25
The sum of the first four terms is 11/3, so we have:
a + ar + ar^2 + ar^3 = 11/3
Substitute the first equation into the second equation:
25 + ar^2 + ar^3 = 11/3
Multiply everything by 3 to get rid of the fraction:
75 + 3ar^2 + 3ar^3 = 11
Rearrange the equation:
3ar^2 + 3ar^3 = 11 - 75
3ar^2 + 3ar^3 = -64
Simplify:
ar^2 + ar^3 = -64/3
Factor out an a:
a(r + r^2) = -64/3
Since r > 0, r + r^2 is positive. Therefore, a must be negative to make the left side negative.
Let a = -x:
-x(r + r^2) = -64/3
x(r + r^2) = 64/3
Since x > 0, r + r^2 must be positive.
r + r^2 = 64/3x
r^2 + r - 64/3x = 0
Using the quadratic formula:
r = (-1 + sqrt(1 + 4(64/3)x))/2
r = (-1 - sqrt(1 + 4(64/3)x))/2
Since r > 0, we take the positive root:
r = (-1 + sqrt(1 + 256x))/2
Substitute this back into the equation a + ar = 25:
-x(-1 + sqrt(1 + 256x))/2 = 25
x(1 - sqrt(1 + 256x))/2 = 25
Now, we need to solve for x and subsequently find the values of a and r.