The first three terms of an arithmetic progression are 1/2, x, 25. The first three terms of a geometric progression are x+1/4, 32 1/2, and y, where x and y are positive numbers. Find the value of X and the value of Y.
(25 - 1/2)/2 = 49/4
so, x = 49/4
(32 1/2)/(50/4) = y/(32 1/2)
y = 169/2
1/2, x, 25 are in AP
x - 1/2 = 25-x
2x = 51/2
x = 51/4
x+1/4, 32 1/2, and y are in GP, so
(65/2) / (x + 1/4) = y / (65/2)
y(x+1/4) = 4225/4
y( 51/4 + 1/4) = 4225/4
y( 13) = 4225/4
y = 4224/52= 325/4
check:
for AP: 1/2, 51/4, 25
51/4 - 1/2 = 49/4
25 - 51/4 = 49/4 , x is correct
for GP:
x+1/4, 32 1/2 or 13 , 65/2 , 325/4
(65/2) / 13 = 5/2
(325/4) / (65/2) = 5/2 , the y is correct
I calculated the difference, then erroneously assigned that to x.
My bad.
To solve this problem, we'll use the given information about arithmetic and geometric progressions.
Let's start by finding the value of x in the arithmetic progression.
The formula for an arithmetic progression is:
an = a1 + (n - 1)d
where:
an is the nth term of the sequence
a1 is the first term of the sequence
n is the position of the term in the sequence
d is the common difference between terms
Given that the first term (a1) is 1/2, the second term (x) is unknown, and the third term (25) is known, we can write the following equations:
1/2 = a1 + (2 - 1)d (Equation 1: for the first term)
x = a1 + (3 - 1)d (Equation 2: for the second term)
25 = a1 + (4 - 1)d (Equation 3: for the third term)
Substituting the values from the given progression into equations 1, 2, and 3, we have:
1/2 = 1/2 + d (Equation 1)
x = 1/2 + 2d (Equation 2)
25 = 1/2 + 3d (Equation 3)
Simplifying Equation 1, we get:
1/2 = 1/2 + d
d = 0
Since the common difference (d) is 0, Equation 2 becomes:
x = 1/2 + 2(0)
x = 1/2
So, the value of x is 1/2.
Now, let's find the value of y in the geometric progression.
The formula for a geometric progression is:
an = a1 * r^(n - 1)
where:
an is the nth term of the sequence
a1 is the first term of the sequence
r is the common ratio between terms
Given that the first term (a1) is x + 1/4, the second term (32 1/2) is known, and the third term (y) is unknown, we can write the following equations:
x + 1/4 = a1 * r^(2 - 1) (Equation 4: for the first term)
32 1/2 = a1 * r^(3 - 1) (Equation 5: for the second term)
y = a1 * r^(4 - 1) (Equation 6: for the third term)
Substituting the values from the given progression into equations 4, 5, and 6, we have:
x + 1/4 = (x + 1/4) * r (Equation 4)
32 1/2 = (x + 1/4) * r^2 (Equation 5)
y = (x + 1/4) * r^3 (Equation 6)
Simplifying Equation 4, we get:
x + 1/4 = (x + 1/4) * r
1 = r
Since the common ratio (r) is 1, Equation 5 becomes:
32 1/2 = (x + 1/4) * (1)^2
32 1/2 = (x + 1/4)
Simplifying Equation 6, we get:
y = (x + 1/4) * (1)^3
Substituting the value of x from earlier (x = 1/2), we have:
y = (1/2 + 1/4)
Simplifying y, we get:
y = 3/4
Therefore, the value of x is 1/2 and the value of y is 3/4.