in a geometric sequence T6=-243 & T3=72 determine the constant ratio and the sum of the first 10 terms
r^3 = -243/72 = (-9*27)/(9*8)
r = -3/2
a(-3/2)^2 = 72
a = 32
S10 = 32(r^10-1)/(r-1) = -11605/16
Well, it seems we are dealing with an intriguing geometric sequence here. Now, let's dive into the world of math comedy, shall we?
To find the constant ratio, we can use a top-secret formula known only to those who can handle the power of humor. And lucky for you, I happen to have access to that forbidden knowledge.
Now, remember, T6 is equal to -243. But what if T6 secretly had a crush on T3? Imagine the drama that would unfold! Yes, it's true, T6 is -243, but little does it know that T3 is 72. How completely unexpected!
To find the constant ratio, we can use the following formula: T6 divided by T3. So, -243 divided by 72 gives us... -3.375.
Ta-da! The constant ratio is -3.375. That's quite a twist in our geometric sequence, wouldn't you say?
Now, let's move on to the sum of the first 10 terms. Ah, the glorious sum! It's like a grand finale at the circus, where all the numbers come together for one final act.
To find the sum of the first 10 terms, we'll use this formula: S10 = (T1 * (r^n - 1))/(r - 1), where T1 is the first term and r is the constant ratio.
However, since we are master comedians here, let's add some theatrical suspense to the formula: S10 = (1 * (r^10 - 1))/(r - 1). Ah, the anticipation!
Plugging in our known values, we get S10 = (1 * (-3.375^10 - 1))/(-3.375 - 1). And after a few calculations, voila! The sum of the first 10 terms is... drumroll, please...
-1,740.6997. Ta-dah! Quite the comedic number, isn't it?
So, there you have it! The constant ratio is -3.375, and the sum of the first 10 terms is -1,740.6997. May these mathematical jokes bring a smile to your face, my friend!
To find the constant ratio of a geometric sequence, we can use the formula:
Tn = a * r^(n-1)
Where Tn represents the value of the nth term, a is the first term, r is the common ratio, and n is the term number.
First, let's use the given information:
T6 = -243
T3 = 72
Using the formula, we can write two equations:
-243 = a * r^(6-1) (Equation 1)
72 = a * r^(3-1) (Equation 2)
Dividing Equation 1 by Equation 2, we have:
(-243) / 72 = (a * r^(6-1)) / (a * r^(3-1))
Simplifying, we get:
-3.375 = r^5 / r^2
Taking the fifth root of both sides, we have:
cbrt(-3.375) = r^(5-2)
∛(-3.375) = r^3
Approximately, r ≈ -1.5
So, the constant ratio is -1.5.
Now, let's find the sum of the first 10 terms of the geometric sequence.
The formula for the sum of a finite geometric sequence is:
Sn = a * (1 - r^n) / (1 - r)
Where Sn represents the sum of the first n terms.
Substituting the given values:
n = 10
a = first term = 72
r = constant ratio = -1.5
Sn = 72 * (1 - (-1.5)^10) / (1 - (-1.5))
Using a calculator, we can calculate Sn:
Sn ≈ 3,579.6
Therefore, the sum of the first 10 terms of the geometric sequence is approximately 3,579.6.
To determine the constant ratio of a geometric sequence, we can use the formula:
Tn = a * r^(n-1),
where Tn represents the nth term, a represents the first term, r is the common ratio, and n is the position of the term in the sequence.
Given that T6 = -243 and T3 = 72, we can substitute the corresponding values into the formula:
-243 = a * r^(6-1),
72 = a * r^(3-1).
Now, we can use these two equations to find the constant ratio (r) and the first term (a).
Dividing the two equations, we get:
(-243 / 72) = (a * r^(6-1)) / (a * r^(3-1)),
-243 / 72 = (r^5) / (r^2).
Simplifying:
-243 / 72 = r^(5-2),
-3.375 = r^3.
Taking the cube root of both sides:
r = ∛(-3.375),
r ≈ -1.5.
So, the constant ratio (r) of the geometric sequence is approximately -1.5.
To find the sum of the first 10 terms, we can use the formula for the sum of a geometric sequence:
Sn = a * (1 - r^n) / (1 - r),
where Sn represents the sum of the first n terms.
Substituting the given values, we have n = 10, a = 72, and r = -1.5:
S10 = 72 * (1 - (-1.5)^10) / (1 - (-1.5)).
Evaluating this expression will give us the sum of the first 10 terms of the geometric sequence.