The sum of three consecutive terms of a geometric progression is 42, and their product is 512. Find the three terms.
use your definitions:
"The sum of three consecutive terms of a geometric progression is 42"
----> a + ar + ar^2 = 42
a(1 + r + r^2) = 42 **
"their product is 512" --- a(ar)(ar)^2 = 512
a^3 r^3 = 512
(ar)^3 = 8^3
ar = 8 ***
divide ** by ***
(1+r + r^2)/r = 42/8 = 21/4
4r^2 + 4r + 4 = 21r
4r^2 - 17r + 4 = 0
(4r - 1)(r - 4) = 0
r = 1/4 or r = 4
if r = 4, in ar= 8 , a = 2
the 3 terms are 2, 8, and 32
check: sum = 2+8+32 = 42 ⩗
product = 2 x 8 x 32 = 512 ⩗
I will leave it up to you to find the other case.
Thank you so much for your help,it helps to clear some confusions😃😃
Good method of solving
To find the three terms of the geometric progression, we need to use the given information.
Let's assume the first term of the geometric progression is "a", and the common ratio between the terms is "r".
Since the sum of three consecutive terms is 42, we can set up the following equation:
a + ar + ar^2 = 42 ---(Equation 1)
Similarly, since the product of the three terms is 512, we can set up another equation:
a * ar * ar^2 = 512 ---(Equation 2)
To solve these equations, we can simplify Equation 2 by multiplying the terms:
a * ar * ar^2 = a * (ar) * (ar^2)
= a^3 * (r * r^2)
= a^3 * r^3
So, Equation 2 becomes:
a^3 * r^3 = 512 ---(Equation 3)
Now, we have three equations:
Equation 1: a + ar + ar^2 = 42
Equation 2: a^3 * r^3 = 512
Equation 3: a^3 * r^3 = 512
To solve these equations, we'll first simplify Equation 1 by factoring out "a":
a(1 + r + r^2) = 42
Now, let's rewrite Equation 1 as:
a = 42 / (1 + r + r^2) ---(Equation 4)
Substituting Equation 4 into Equation 3, we get:
(42 / (1 + r + r^2))^3 * r^3 = 512
Cubing both sides of the equation:
(42^3 / (1 + r + r^2)^3) * r^3 = 512
Multiply both sides by (1 + r + r^2)^3:
(42^3 * r^3) = 512 * (1 + r + r^2)^3
Now, we can simplify and solve for "r":
r^3 = (512 * (1 + r + r^2)^3) / (42^3)
Taking the cube root of both sides:
r = CubeRoot[(512 * (1 + r + r^2)^3) / (42^3 )]
After finding the value of "r", substitute it back into Equation 4 to find the value of "a".
Finally, we can find the three terms by evaluating:
First term = "a"
Second term = "a * r"
Third term = "a * r^2"