In Geometric progression the sum of first of three terms is 28 and the product is 512 find those numbers
a+ar+ar^2=28
a^3 * r^3=512
one the second equation, take the cube root of each side.
ar=cubrt(512)=8
a+8+ar^2=28
a(1+r^2)=20
but a= 8/r
8(1+r^2)=20r
8r^2-20 r+8=0
r=(20)+-sqrt(400-256) /16
r=20+- sqrt(144) /16 = (20+-12)/16=2 or 1/2
then a= 4 or 16
check a=4, r=2
4 + 8 + 16=28
a^3 r^3=64*8=512 checks, a=4, r=2
check a=8, r=1/2
8 +4 + 2 does not check.
Let's denote the first term of the geometric progression as "a" and the common ratio as "r".
The sum of the first three terms of a geometric progression can be given by the formula:
S = a + ar + ar^2
Substituting the given values, we have:
28 = a + ar + ar^2 ------(1)
The product of the three terms is given by:
P = a * ar * ar^2
Substituting the given values, we have:
512 = a * ar * ar^2 ------(2)
Now, let's solve these equations step by step.
From equation (1), we can express "a" as:
a = 28 - ar - ar^2
Substituting this expression for "a" into equation (2), we have:
512 = (28 - ar - ar^2) * ar * ar^2
Expanding the expressions and simplifying, we get:
512 = 784r^3 - 28r^2 - 28ar^3 + ar^4
Rearranging the terms, we have:
ar^4 + (28 - 28r^3) * a - 28r^2 + 784r^3 - 512 = 0
This is a quadratic equation in terms of "a". Rearranging it, we get:
ar^4 - ar(28r^3) + 784r^3 - 28r^2 - 512 = 0
Now, we can use a numerical method or factoring to solve this equation. Let's assume r = 2:
(2a)^4 - (2a)(2^3 * 28) + 784(2^3) - 28(2^2) - 512 = 0
16a^4 - 448a + 1792 - 112 + 1792 - 112 - 512 = 0
16a^4 - 448a + 2752 - 624 = 0
16a^4 - 448a + 2128 = 0
We can simplify further by dividing by 16:
a^4 - 28a + 133 = 0
Unfortunately, the equation does not have any rational solutions. Hence, we need to use numerical methods such as approximation or calculators to find the values of "a" and "r". These methods involve guesswork and calculations beyond the capabilities of this text-based interface.
Please note that this is an example of how to set up the problem and solve it step-by-step. The actual values of "a" and "r" likely involve more complex calculations.
To find the numbers in a geometric progression, we'll start by assigning variables to them. Let's represent the common ratio as 'r' and the first term as 'a'.
The three terms in the geometric progression are:
Term 1: a
Term 2: ar (by multiplying the first term by the common ratio)
Term 3: ar^2 (by multiplying the second term by the common ratio)
Given that the sum of the first three terms is 28, we can write the equation:
a + ar + ar^2 = 28 (equation 1)
The product of the three terms is 512, so we can write another equation:
a * ar * ar^2 = 512 (equation 2)
Now, let's solve these equations simultaneously to find the values of 'a' and 'r'.
From equation 1, we can factor out 'a':
a(1 + r + r^2) = 28
From equation 2, we can simplify:
a^3 * r^3 = 512
Taking the cube root of both sides of equation 2, we get:
ar = 8 (cube root of 512 is 8)
Substituting this value of 'ar' into equation 1, we have:
a(1 + 8) = 28
9a = 28
Dividing both sides by 9, we find:
a = 28/9
Now that we have the value of 'a', we can substitute it back into equation 2 to solve for 'r':
(28/9)^3 * r^3 = 512
Taking the cube root of both sides, we get:
r^3 = 512 / (28/9)^3
Simplifying the right side, we have:
r^3 = 512 * (9/28)^3
Taking the cube root of both sides, we find:
r = (512 * (9/28)^3)^(1/3)
Therefore, by calculating 'a' and 'r', we can determine the three terms in the geometric progression.