the sum of the first four terms of a geometry series is 15 and the sum of the next 4 terms is 240.determine the positive constant ratio.
terms are
a , ar, ar^2 ...
sum4 = a(r^4 - 1)/(r-1) = 15
sum8 = a(r^8 - 1)/(r-1) = 15 + 240 = 255
divide the 2nd equation by the 1st
(r^8-1)/(r^4 - 1) = 255/15 = 17
r^4+1)(r^4-1)/(r^4-1) = 17
r^4+1 = 17
r^4 = 16
r^2 = ±4 , and for reals
r = ±2
so the constant ratio is either +2 or -2
Well, well, well, looks like we're dealing with a sneaky geometric series here. Let's clown around and figure it out!
The sum of the first four terms of the series is 15, huh? That means we need to find a constant ratio that makes those terms add up to 15. Let's call that constant ratio "r".
So, our first term is "a", the second term is "ar", the third term is "ar^2", and the fourth term is "ar^3".
If we add them all up, we get:
a + ar + ar^2 + ar^3 = 15
Now, the sum of the next four terms is 240. That means we need to find a constant ratio that makes those terms add up to 240. Let's call that constant ratio "r" again.
So, our fifth term is "ar^4", the sixth term is "ar^5", the seventh term is "ar^6", and the eighth term is "ar^7".
If we add them all up, we get:
ar^4 + ar^5 + ar^6 + ar^7 = 240
Now, let's put on our clown shoes and do some math!
Divide the sum of the second equation by the sum of the first equation:
(ar^4 + ar^5 + ar^6 + ar^7) / (a + ar + ar^2 + ar^3) = 240 / 15
Simplify it:
r^4 + r^5 + r^6 + r^7 = 16
Now, now, don't panic! We're getting closer to revealing that constant ratio.
Divide every term by r^4 to simplify things:
1 + r + r^2 + r^3 = 16 / r^4
Hmmm, this isn't looking very clownish at the moment. Let's see if we can make it more fun!
Notice anything about the left side of the equation? It's the sum of a geometric series! Look at you, paying attention.
Now, if we know anything about sum formulas for geometric series, we know that:
1 + r + r^2 + r^3 = (r^4 - 1) / (r - 1)
So let's replace that left side with our lovely sum formula:
(r^4 - 1) / (r - 1) = 16 / r^4
Cross-multiply because we like to play around:
r^5 - r^4 = 16 - r^4
Combine like terms:
r^5 = 16
And now, ladies and gentlemen, we have found our constant ratio! The positive constant ratio for this geometric series is 16. Ta-da!
I hope you enjoyed my clownish approach to solving this problem. Remember, learning can be fun too!
To find the positive constant ratio, we can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r),
where:
- S is the sum of the series,
- a is the first term,
- r is the common ratio, and
- n is the number of terms.
Let's denote the first term as a and the common ratio as r. We are given the sum of the first four terms as 15, so we have:
15 = a * (1 - r^4) / (1 - r)^1. (Equation 1)
We are also given the sum of the next four terms as 240, so we have:
240 = a * (1 - r^8) / (1 - r). (Equation 2)
To solve for the value of r, we can divide Equation 2 by Equation 1:
240 / 15 = (a * (1 - r^8) / (1 - r)) / (a * (1 - r^4) / (1 - r)).
Simplifying and canceling out the common terms on both sides:
16 = (1 - r^8) / (1 - r^4).
Now, let's cross multiply:
(1 - r^8) * (1 - r) = 16 * (1 - r^4).
Expanding both sides:
(1 - r - r^8 + r^9) = 16 - 16r^4.
Bringing all the terms to one side:
16r^4 - r^9 + r^8 - r + 15 = 0. (Equation 3)
Now, we need to solve Equation 3 to find the value of r. However, this equation might require numerical methods to find a precise answer since it is a polynomial equation of degree 9.
I hope this helps! Let me know if you have any further questions.
To determine the positive constant ratio of the geometric series, we need to set up a system of equations based on the given information.
Let's assume that the first term of the geometric series is 'a' and the common ratio is 'r'.
The sum of the first four terms can be expressed as:
S_4 = a + ar + ar^2 + ar^3
The sum of the next four terms can be expressed as:
S_8 = a + ar + ar^2 + ar^3 + ar^4 + ar^5 + ar^6 + ar^7
We are given that the sum of the first four terms (S_4) is 15:
S_4 = 15
And the sum of the next four terms (S_8) is 240:
S_8 = 240
Now we can substitute the formulas for S_4 and S_8:
15 = a + ar + ar^2 + ar^3 (Equation 1)
240 = a + ar + ar^2 + ar^3 + ar^4 + ar^5 + ar^6 + ar^7 (Equation 2)
To eliminate 'a' from the equations, we can subtract Equation 1 from Equation 2:
225 = ar^4 + ar^5 + ar^6 + ar^7
Next, we can factor out 'ar^4' from the right side of the equation:
225 = ar^4(1 + r + r^2 + r^3)
Since 'r' is a positive constant, we can rewrite the equation as:
225 = ar^4 * (1 + r + r^2 + r^3) (Equation 3)
Now, we have two equations:
Equation 1: 15 = a + ar + ar^2 + ar^3
Equation 3: 225 = ar^4 * (1 + r + r^2 + r^3)
To solve this system of equations, we can divide Equation 3 by Equation 1:
225 / 15 = (ar^4 * (1 + r + r^2 + r^3)) / (a + ar + ar^2 + ar^3)
Simplifying the equation:
15 = r^4 * (1 + r + r^2 + r^3) / (1 + r + r^2 + r^3)
Since 'r' is a positive constant, we can cancel out the common factors:
15 = r^4
Now, taking the fourth root of both sides:
∛15 = ∛(r^4)
So, the positive constant ratio 'r' of the geometric series is equal to the cube root of 15.
Therefore, the positive constant ratio is ∛15 (approximately 2.466).