in a geometric sequence T6=-243 & T3=72 determine the constant ratio
Well, well, well, it seems we have a geometric sequence mystery on our hands! Let's solve it with a touch of humor, shall we?
First things first, we have T3 = 72 and T6 = -243. To find the constant ratio (r) between the terms, we can use a little equation I like to call "the divide-and-conquer method." Get it? Because we're dividing the terms and conquering the constant ratio? Anyway, let's move on.
The formula for the nth term of a geometric sequence is Tn = a * r^(n-1), where a is the first term and r is the constant ratio. In our case, we have:
T3 = a * r^(3-1) = 72
T6 = a * r^(6-1) = -243
Now, we can divide T6 by T3 and see what happens:
(-243) / 72 = (a * r^(6-1)) / (a * r^(3-1))
-3.375 = r^5 / r^2
-3.375 = r^(5-2)
-3.375 = r^3
Aha! We've isolated r to the power of 3, but we still need r itself. So, let's take the cube root of both sides of our equation:
∛(-3.375) = ∛(r^3)
-1.5 = r
And there you have it! The constant ratio of this geometric sequence is -1.5. Don't worry, though – this sequence won't leave you feeling negative. It just adds a little twist to the progression!
To find the constant ratio of a geometric sequence, you can use the formula:
\[
T_n = T_1 \times r^{(n-1)}
\]
where:
- \(T_n\) represents the \(n\)th term of the sequence,
- \(T_1\) represents the first term of the sequence, and
- \(r\) represents the common ratio of the sequence.
Given that \(T_6 = -243\) and \(T_3 = 72\), we can plug in the values into the formula to form two equations:
Equation 1:
\[
-243 = T_1 \times r^{(6-1)}
\]
Equation 2:
\[
72 = T_1 \times r^{(3-1)}
\]
Dividing Equation 1 by Equation 2, we get:
\[
\frac{-243}{72} = \frac{T_1 \times r^{(6-1)}}{T_1 \times r^{(3-1)}}
\]
Simplifying, we have:
\[
-3.375 = r^{(6-1)-(3-1)}
\]
\[
-3.375 = r^{5-2}
\]
\[
-3.375 = r^3
\]
To find the value of \(r\), we need to take the cube root of both sides of the equation:
\[
r = \sqrt[3]{-3.375}
\]
Using a calculator, we can evaluate the cube root of -3.375:
\[
r \approx -1.5
\]
Therefore, the constant ratio of the geometric sequence is approximately -1.5.
To determine the constant ratio in a geometric sequence, we need to use the given terms of the sequence.
We are given:
T6 = -243
T3 = 72
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. Let's denote this constant ratio as "r".
Now, we can use the given terms to set up two equations:
T6 = T3 * r^3
-243 = 72 * r^3 (Substituting the given values)
We can divide both sides of the equation by 72 to simplify it:
-243/72 = r^3
Simplifying further:
-3.375 = r^3
To find the value of "r", we can take the cube root of both sides of the equation:
∛(-3.375) = ∛(r^3)
r = -1.5
Therefore, the constant ratio in this geometric sequence is -1.5.