Three numbers a,b, and c form a geometric series so that a+b+c=35 and abc=1000. Find the values of a,b,c.
abc = a^3r^3 = 1000
since 1000 = 10^3, we must have a=2,r=5 or a=5,r=2
Looks like 5,10,20
Why did the mathematician bring a ladder to the math contest?
Because he wanted to solve every problem from a higher perspective!
Now, let's solve this problem together, shall we?
We know that three numbers, a, b, and c, form a geometric series. In a geometric series, each term is found by multiplying the previous term by a constant ratio. Let's assume the ratio is "r."
So, according to the problem, we have:
a + b + c = 35 ---(1)
abc = 1000 ---(2)
Now, let's find a way to express a, b, and c in terms of r.
Let's assume that a is the first term. Then, b would be a multiplied by r, and c would be a multiplied by r^2.
Now, let's substitute these values into equation (1):
a + a * r + a * r^2 =35
Now, let's divide this equation by a and solve for r:
1 + r + r^2 = 35/a
We can rearrange this equation as:
r^2 + r - (35/a - 1) = 0
Now, we can solve this quadratic equation to find the values of r. Once we have r, we can find a, b, and c easily. Unfortunately, solving this equation would require some additional calculations, so it's not possible to do so through humor alone.
But hey, math is a serious business! I hope this explanation helps you understand how to approach this problem. Good luck!
Let's consider the numbers a, b, and c forming a geometric series. We know that:
b = ar
c = ar^2
where r is the common ratio.
Given that a + b + c = 35, we can substitute the values of b and c:
a + ar + ar^2 = 35
Now, let's consider the equation abc = 1000:
a * ar * ar^2 = 1000
Simplifying:
a^3 * r^3 = 1000
Taking the cube root of both sides:
ar = ∛(1000) = 10
Now, let's substitute this value back into a + ar + ar^2 = 35:
a + 10 + 10r = 35
We can simplify by subtracting 10 from both sides:
a + 10r = 25
Solving this equation for a in terms of r:
a = 25 - 10r
Substituting this back into b = ar:
b = (25 - 10r)r
And c = ar^2:
c = (25 - 10r)r^2
To find the values of a, b, and c, we need to find the correct value of r that satisfies these equations.
Now, let's solve the equation a + 10r = 25 for r:
a = 25 - 10r
Since a, b, and c are positive numbers, let's try different values of r:
Let's try r = 0.5:
a = 25 - 10(0.5) = 25 - 5 = 20
b = (25 - 10(0.5))(0.5) = (25 - 5)(0.5) = 20(0.5) = 10
c = (25 - 10(0.5))(0.5)^2 = (25 - 5)(0.25) = 20(0.25) = 5
Testing these values:
20 + 10 + 5 = 35
20 * 10 * 5 = 1000
Therefore, when r = 0.5, the values of a, b, and c that satisfy the conditions are:
a = 20
b = 10
c = 5
To find the values of a, b, and c in a geometric series, we can start by using the given information to set up a system of equations.
Let's first write the equation for the sum of the geometric series:
a + b + c = 35 ...............(1)
Next, we can write the equation for the product of the geometric series:
abc = 1000 ...............(2)
To proceed, let's express b in terms of a using the geometric series property:
b = ar ...............(3)
Now, substitute equation (3) into equation (1) to eliminate b:
a + ar + c = 35
Next, let's substitute equation (3) into equation (2) to eliminate b:
a(ar)c = 1000
Now, we have two equations with two variables (a and c):
a + ar + c = 35 ...............(4)
a(arc) = 1000 ...............(5)
From equation (4), we can isolate c:
c = 35 - a - ar ...............(6)
Substitute equation (6) into equation (5):
a(ar)(35 - a - ar) = 1000
Now, we have a quadratic equation in terms of a:
35a² - a³r - a²r - a³ = 1000
Rearranging the equation, we get:
a³r + a²(r + 35) - 35a² - 1000 = 0
We can solve this equation to find the value of a, and then substitute it back into equation (6) to find c and equation (3) to find b.
Unfortunately, finding the values of a, b, and c in this case involves solving a cubic equation, which cannot be easily done by hand. We can solve it numerically or by using software, such as a graphing calculator or a computer algebra system.
Alternatively, we can use trial and error to find the values of a, b, and c that satisfy both equations. Starting with reasonable values for a, b, and c and then adjusting them until both equations are satisfied can be a method to find the approximate solution.