The sum of the first three term in a G.P is 38. Their product is 1728. Find the values of the three terms.
a(1+r+r^2) = 38
a*ar*ar^2 = 1728
now divide to get
r^3/(1+r+r^2) = 1728/38
now solve for r, then get a
note that (ar)^3 = 1728 = 12^3
so the 2nd term, ar = 12
the 1st term is 12/r
the 3rd term is 12r
Let's denote the first term as 'a' and the common ratio as 'r' in the geometric progression (G.P).
The sum of the first three terms of a G.P can be found using the formula:
S3 = a + a*r + a*r^2
Given that S3 = 38, we can substitute the values:
38 = a + a*r + a*r^2 ........(1)
The product of the three terms can be found using the formula:
P3 = a * (a*r) * (a*r^2)
Given that P3 = 1728, we can substitute the values:
1728 = a * (a*r) * (a*r^2)
1728 = a^3 * r^3 ........(2)
Now, we have two equations (1) and (2) with two variables (a and r). We can solve these equations simultaneously to find the values of 'a' and 'r'.
From equation (1), we can rewrite it as:
38 = a * (1 + r + r^2)
Dividing equation (2) by equation (1):
1728 / 38 = a^3 * r^3 / (a * (1 + r + r^2))
45.47 = a^2 * r^2
Since both sides of the equation are squares, we can take the square root:
√(a^2 * r^2) = √45.47
a * r = √45.47
Now, we have two equations:
1. 38 = a * (1 + r + r^2)
2. a * r = √45.47
Now, we can solve these two equations to find the values of 'a' and 'r'.
Sure! To find the values of the three terms in a geometric progression (G.P), we can use the formulas involving the sum and product of terms in a G.P.
Let's suppose the first term of the G.P is 'a' and the common ratio is 'r'.
According to the given information, the sum of the first three terms is 38. In a G.P, the sum of the first three terms can be calculated using the formula:
Sum of first three terms = a + ar + ar^2
Given that the sum is 38, we have the equation:
38 = a + ar + ar^2 ---(1)
The product of the three terms is 1728. In a G.P, the product of the first three terms can be calculated using the formula:
Product of first three terms = a * ar * ar^2
Given that the product is 1728, we have the equation:
1728 = a * ar * ar^2 ---(2)
Now, we have a system of two equations (equation 1 and equation 2) with two variables (a and r). We can solve these equations simultaneously to find the values of 'a' and 'r'.
To solve the system of equations, we can substitute the value of 'a' from equation 1 into equation 2 or vice versa. Then we can solve for 'r' and use that value to find 'a'.
Let's solve the system of equations step by step:
From equation 1, we can rearrange the terms and express 'a' in terms of 'r':
a = 38 - ar - ar^2 ---(3)
Now, substitute the value of 'a' from equation 3 into equation 2:
1728 = (38 - ar - ar^2) * (ar) * (ar^2)
Expanding this equation, we get:
1728 = 38ar^3 - 38ar^2 - 38ar^4 + a^2r^3
Now, rearrange everything to one side of the equation:
38ar^4 + 38ar^3 + 38ar^2 - a^2r^3 - 1728 = 0
Now, we have a fourth-degree polynomial equation in terms of 'r'. We can use numerical or symbolic methods to solve this equation.
By solving this equation, we will find the values of 'r'. Once we have 'r', we can substitute it back into equation 3 to find the value of 'a'.
So, to find the values of the three terms in the geometric progression, you'll need to solve the fourth-degree polynomial equation. You can use numerical methods like the Newton-Raphson method or symbolic methods like factoring or using computer software to find the values of 'r'.
Once you find the value of 'r', you can substitute it back into equation 3 to find the value of 'a'.