the third terms of a geometric progression is 360 and the sixth term is 1215. find the sum of the first four terms....
Explain well
How did you get 27/8
To find the sum of the first four terms of a geometric progression, we first need to determine the common ratio (r) of the progression.
Let's call the first term of the geometric progression "a" and the common ratio "r".
Given the information that the third term is 360, we can write the following equation:
a * r^2 = 360 (equation 1)
Similarly, we can write the equation for the sixth term being 1215:
a * r^5 = 1215 (equation 2)
To solve for "a" and "r," we can divide equation 2 by equation 1:
(a * r^5) / (a * r^2) = 1215 / 360
Simplifying, we get:
r^3 = 1215 / 360
Now, solve for r by taking the cube root of both sides:
r = (1215 / 360)^(1/3)
Using a calculator, we find that r ≈ 1.5
Now that we know r, let's find the first term "a" by substituting back into equation 1:
a * (1.5)^2 = 360
a * 2.25 = 360
Divide both sides by 2.25:
a = 360 / 2.25
a ≈ 160
Now we have the first term "a" ≈ 160 and the common ratio "r" ≈ 1.5.
To find the sum of the first four terms, we can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Plugging in the values, we get:
S = 160 * (1 - 1.5^4) / (1 - 1.5)
S = 160 * (1 - 5.0625) / (-0.5)
S ≈ -160 * (-4.0625) / 0.5
S ≈ 1290
ar^5 = 1215
ar^2 = 360
now divide, and you have
r^3 = 27/8
you can take it from there, right?