if 1,x,y is a geometric sequence and x,y,3 is an arithmetic sequence, compute the maximum value of x+y
y/x = x, so
y = x^2
3-y = y-x, so
2y = x+3
2x^2 = x+3
2x^2 - x - 3 = 0
(x+1)(2x-3) = 0
x = -1 or 3/2
y = 1 or 9/4
x+y = 0 or 15/4
Go with Steve's
I was standing on my head when I got
2x-3 = 0 ----> x = 2/3
To find the maximum value of x + y, we need to determine the values of x and y that satisfy the given conditions.
Let's go step by step:
Step 1: The sequence 1, x, y is a geometric sequence.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio. Let's represent this common ratio as r.
So, the terms in the sequence 1, x, y can be written as follows:
First term: 1
Second term: 1 * r = x
Third term: x * r = y
Step 2: The sequence x, y, 3 is an arithmetic sequence.
In an arithmetic sequence, each term is obtained by adding a constant called the common difference to the previous term. Let's represent this common difference as d.
So, the terms in the sequence x, y, 3 can be written as follows:
First term: x
Second term: x + d = y
Third term: y + d = 3
Step 3: Solving the equations:
From the geometric sequence, we know that:
1 * r = x -----(Equation 1)
x * r = y -----(Equation 2)
From the arithmetic sequence, we know that:
x + d = y -----(Equation 3)
y + d = 3 -----(Equation 4)
Let's solve these equations:
From Equation 1, we can rewrite it as:
r = x/1
Substituting this value of r into Equation 2, we get:
(x/1) * x = y
x^2 = y -----(Equation 5)
Substituting the value of y from Equation 3 into Equation 5, we get:
x^2 = x + d
Substituting the value of y from Equation 4 into Equation 3, we get:
x + d = 3 - d
2d = 3 - x
Now, we can substitute the value of x + d from Equation 3 into Equation 5, and the value of x + d from Equation 4 into Equation 3:
x^2 = 3 - d
2d = 3 - x
Rearranging Equation 1, we get:
x = r
Substituting this value of x and Equation 2 into Equation 3, we get:
r + d = r^2
Now, we have a system of equations:
x^2 = 3 - d
r + d = r^2
Step 4: Solving the system of equations:
To find the maximum value of x + y, we need to maximize the sum of x and y. This is equivalent to maximizing the value of x + d since y = x + d.
Let's solve the system of equations using the given conditions:
From Equation 2, we can rewrite it as:
d = r^2 - r
Substituting this value of d into Equation 1, we get:
x^2 = 3 - (r^2 - r)
x^2 = 3 - r^2 + r
x^2 + r^2 - r = 3 -----(Equation 6)
Substituting the value of d from Equation 2 into Equation 4, we get:
r + (r^2 - r) = 3
r^2 = 2
To maximize the value of x + d (or x + y), we need to find the largest possible value of x.
Substituting the value of r^2 from above into Equation 6, we get:
x^2 + 2 - r = 3
x^2 = 1 + r
Substituting the value of r^2 from above into Equation 4, we get:
2 + (r^2 - r) = 3
r^2 - r = 1
Now we have a system of equations:
x^2 = 1 + r
r^2 - r = 1
Step 5: Solving the system of equations:
From Equation 7, we can express r as a function of x by rearranging Equation 5:
r = 1 + x^2
Substituting this value of r into Equation 6, we get:
(1 + x^2)^2 - (1 + x^2) = 1
Expanding and simplifying, we get:
1 + 2x^2 + x^4 - 1 - x^2 = 1
x^4 + x^2 = 1
Multiplying through by x^2, we get:
x^6 + x^4 - x^2 = 0
Now, we have a polynomial equation of degree 6. To find the roots of this equation, we can use numerical methods or factorization techniques. However, solving this equation exactly can be quite complex.
Alternatively, you could use a graphing calculator or software to plot the graph of the equation and find the x-values at which the curve intersects the x-axis. Then, these x-values can be substituted into the equations to find the corresponding values of y.
Finally, once you have all the possible values of x and y, you can compute x + y for each pair and determine the maximum value.
Note: This explanation provides the general procedure to solve the problem. However, due to the complexity of the equations involved, finding the exact solutions may require further mathematical techniques or numerical analysis methods.
from the GS:
x/1 = y/x
y = x^2
from the AS:
y-x= 3-y
2y = x+3
y = (x+3)/2
x^2 = (x+3)/2
2x^2 - x - 3 = 0
2x - 3)(x + 1) = 0
x = 2/3 or x = -1
if x = 2/3, then y = 4/9 and
x+y = 2/3 +4/9 = 10/9
if x = -1, then y = 1 and
x+y = 0
so the maximium value of x+y = 10/9