the third term of gp is 360 and the sixth term is 1215 find the common ratio
Tn = a r^(n-1)
T3 = a r^2 = 360
T6 = a r^5 = 1215
so
r^5/r^2 = 1215/360 = 3.375
r = 3.375^(1/3) = 1.5
Well, if the third term of a geometric progression is 360 and the sixth term is 1215, then it's clear that the common ratio is not paying its dues! It needs to step up its game and make a visible difference. But fear not, I'll help you figure it out.
To find the common ratio (or "r" as mathematics prefers to call it), we need to find the ratio between consecutive terms.
If we divide the sixth term (1215) by the third term (360), we should get the common ratio. So, let's see:
1215 divided by 360 equals 3.375.
Voila! The common ratio is 3.375. It may not be the most straightforward or popular number, but hey, it does the job!
To find the common ratio, we can use the formula for the nth term of a geometric progression (GP):
Tn = a * r^(n-1)
where Tn is the nth term, a is the first term, r is the common ratio, and n is the term number.
Given:
Third term, T3 = 360
Sixth term, T6 = 1215
Let's start by finding the first term (a).
We can substitute the values for T3 and T6 into the formula to form two equations:
T3 = a * r^(3-1) ---> Equation 1
T6 = a * r^(6-1) ---> Equation 2
Substituting the given values:
360 = a * r^2 ---> Equation 1
1215 = a * r^5 ---> Equation 2
To solve these equations, divide Equation 2 by Equation 1:
(1215 / 360) = (a * r^5) / (a * r^2)
3.375 = r^3
Now, we can take the cube root of both sides to solve for r:
r = (∛3.375) ≈ 1.5
Therefore, the common ratio of the geometric progression is approximately 1.5.
To find the common ratio (r) of a geometric progression (GP), we can use the formula:
nth term = a * r^(n-1)
where a is the first term, n is the position of the term in the sequence, and r is the common ratio.
Given that the third term (n = 3) is 360 and the sixth term (n = 6) is 1215, we can set up the following equations:
360 = a * r^(3-1)
1215 = a * r^(6-1)
To solve for the common ratio, we can divide the second equation by the first equation:
1215 / 360 = (a * r^(6-1)) / (a * r^(3-1))
Simplifying, we get:
3.375 = r^5 / r^2
To simplify further, we can divide r^5 by r^2, which gives us:
3.375 = r^(5-2)
3.375 = r^3
To solve for r, we can take the cube root of both sides:
∛3.375 = ∛r^3
1.5 = r
Therefore, the common ratio (r) of the geometric progression is 1.5.