The sum of the 3rd term & 5th term of a GP is -60 and the sum of the 5th & 7th term is -240.find the common ratio
ar^2 + ar^4 = -60
ar^4 + ar^6 = -240
ar^2 (1+r^2) = -60
ar^4 (1+r^2) = -240
-60/ar^2 = -240/ar^4
-60r^2 = -240
r = ±2
To find the common ratio (r) of a geometric progression (GP) given the sum of certain terms, we can set up a system of equations using the given information.
Let's denote the first term of the GP as 'a' and the common ratio as 'r'.
According to the problem, the sum of the 3rd term and the 5th term of the GP is -60. This can be expressed as:
a * r^2 + a * r^4 = -60 ...........(1)
Similarly, the sum of the 5th and 7th terms of the GP is -240, which can be expressed as:
a * r^4 + a * r^6 = -240 ...........(2)
Now, we have a system of equations with two unknowns, 'a' and 'r'. To solve this, we can use the method of substitution or elimination.
Let's choose the method of substitution. We'll rewrite equation (1) to express 'a' in terms of 'r' and substitute it into equation (2).
From equation (1), we can rearrange to get:
a * r^2 + a * r^4 = -60
a(1 + r^2 * r^2) = -60
a(1 + r^4) = -60
a = -60 / (1 + r^4) ...........(3)
Now, substitute equation (3) into equation (2):
(-60 / (1 + r^4)) * r^4 + (-60 / (1 + r^4)) * r^6 = -240
Multiplying through by (1 + r^4) to eliminate the fraction:
-60 * r^4 + (-60) * r^6 = -240 * (1 + r^4)
-60 * r^4 - 240 - 60 * r^6 = -240 - 240 * r^4
Simplifying:
-60 * r^4 + 60 * r^4 * r^2 + 240 = 240 * r^4
-60 * r^4 + 60 * r^6 + 240 = 240 * r^4
Rearranging the terms:
60 * r^6 + (240 - 240 * r^4) = 60 * r^4
60 * r^6 - 240 * r^4 + (240 - 240) = 0
60 * r^4 * (r^2 - 4) = 0
Now, we have two possibilities:
1. 60 * r^4 = 0
This implies r^4 = 0, which means r = 0.
2. r^2 - 4 = 0
This implies r^2 = 4, which means r = ±2.
So, the possible values for the common ratio (r) are 0, 2, and -2.