If you mean "The sum of the 3rd term and the 5th term of a G.P is -60"
then we have
ar^2 + ar^4 = -60
ar^4 + ar^6 = -240
Dividing, we have
r^2 = 4
Now finish it off
then we have
ar^2 + ar^4 = -60
ar^4 + ar^6 = -240
Dividing, we have
r^2 = 4
Now finish it off
Let's assume that the first term of the G.P. is "a" and the common ratio is "r". We have two pieces of information:
1) The sum of the 3rd term and 5th term is -60:
The 3rd term would be a * r^2, and the 5th term would be a * r^4. So, we can form the equation: a * r^2 + a * r^4 = -60.
2) The sum of the 5th term and 7th term is -240:
The 5th term is a * r^4, and the 7th term is a * r^6. So, we get: a * r^4 + a * r^6 = -240.
Now, get ready for some funny math!
To solve these equations, we can factor out "a" first:
a * (r^2 + r^4) = -60, and a * (r^4 + r^6) = -240.
Since the two equations involve the same quantity (a), we can divide them to eliminate "a":
(r^2 + r^4) / (r^4 + r^6) = -60 / -240.
Simplifying that, we get: (r^2 + r^4) / (r^4 + r^6) = 1/4.
Now, let's keep solving systematically by cross-multiplying:
4(r^2 + r^4) = r^4 + r^6.
Expanding that, we get: 4r^2 + 4r^4 = r^4 + r^6.
Moving all the terms to one side of the equation, we get this hilarious polynomial:
r^6 - 3r^4 + 4r^2 = 0.
Factoring it out, we have: r^2(r^4 - 3r^2 + 4) = 0.
Now, using the Zero Product Property, we can set each factor equal to zero:
1) r^2 = 0: This means r = 0.
2) r^4 - 3r^2 + 4 = 0: This equation is a bit trickier, no pun intended. We'll need to solve it separately.
Unfortunately, the fun stops here. I'm afraid I can't solve this equation for you. You might need the help of a math genius or a magician! Good luck!
1. The sum of the 3rd term and the sum of the 5th term of the G.P is -60:
The sum of an n-term G.P formula is S_n = a(1 - r^n) / (1 - r), where S_n is the sum of the first n terms, a is the first term, and r is the common ratio.
So, we can write the given equation as:
a(1 - r^3) / (1 - r) + a(1 - r^5) / (1 - r) = -60
Simplifying the equation, we get:
a(1 - r^3 + 1 - r^5) / (1 - r) = -60
2a(1 - r^3 - r^5) / (1 - r) = -60
2a(1 - r^3 - r^5) = -60(1 - r)
2. The sum of the 5th term and 7th term of the G.P is -240:
Using the same formula S_n = a(1 - r^n) / (1 - r), we can write the equation:
a(1 - r^5) / (1 - r) + a(1 - r^7) / (1 - r) = -240
Simplifying the equation:
a(1 - r^5 + 1 - r^7) / (1 - r) = -240
2a(1 - r^5 - r^7) = -240(1 - r)
Now we have two equations to work with. We can solve this system of equations to find the values of r and a.
By dividing the second equation by 2 and equating both equations, we get:
2a(1 - r^3 - r^5) = 2a(1 - r^5 - r^7)
-60(1 - r) = -240(1 - r)
-60 + 60r = -240 + 240r
180 = 180r
r = 1
Now substitute the value of r back into one of the original equations to find the value of a.
2a(1 - 1^3 - 1^5) = -60(1 - 1)
2a(0) = 0
a = Any real number
Therefore, the common ratio (r) is 1, and the first term (a) can be any real number.