if 8, x, y, and 27 are four consecutive terms of a geometric progression, find the value of x and y.
27 = 8r^3
so, r = 3/2
Now you can find x and y
I don’t understand how the person solved it it should be step by step which page is dis koraa
To find the values of x and y in the geometric progression, we can use the formula for geometric progression:
An = A1 * r^(n-1)
where An is the nth term, A1 is the first term, r is the common ratio, and n is the position of the term in the sequence.
Given that the terms are 8, x, y, and 27, we can set up two equations using this formula:
x = 8 * r^(2-1)
y = 8 * r^(3-1)
Simplifying the equations, we have:
x = 8 * r
y = 8 * r^2
Since we know that 27 is the fourth term in the progression, we can substitute it into the equation for y:
27 = 8 * r^2
Now, we have a system of two equations:
x = 8 * r
27 = 8 * r^2
To solve the system, we can use substitution or elimination.
Let's solve it using substitution:
From the equation x = 8 * r, we can express r in terms of x:
r = x/8
Substituting this value of r into the equation 27 = 8 * r^2, we get:
27 = 8 * (x/8)^2
27 = x^2/1
x^2 = 27
Taking the square root of both sides, we get:
x = ±√27
Since x needs to be positive (as it is a term in the progression), we take the positive square root:
x = √27
Simplifying, we have:
x = 3√3
To find the value of y, we substitute the value of x back into the equation y = 8 * r^2:
y = 8 * (3√3/8)^2
y = 8 * 9/64
y = 72/64
y = 9/8
Therefore, the value of x is 3√3 and the value of y is 9/8 in the geometric progression.
To find the value of x and y in the geometric progression, we can use the formula for finding the terms of a geometric progression.
In a geometric progression, each term is obtained by multiplying the previous term by a common ratio (r).
Given that 8, x, y, and 27 are consecutive terms in a geometric progression, we can write the following equations:
x = 8 * r (since x is the second term obtained from multiplying 8 by the common ratio r)
y = x * r (since y is the third term obtained from multiplying x by the common ratio r)
27 = y * r (since 27 is the fourth term obtained from multiplying y by the common ratio r)
Now, let's substitute the value of x in terms of r into the equation for y, and then substitute the value of y in terms of r into the equation for 27:
y = (8 * r) * r = 8r^2
27 = (8r^2) * r = 8r^3
Now, we can solve the equation 8r^3 = 27 to find the value of r.
Divide both sides of the equation by 8:
r^3 = 27/8
Take the cube root of both sides to isolate r:
r = ∛(27/8) = 3/2
Now that we know the value of r, we can substitute it back into the equations to find the values of x and y:
x = 8 * (3/2) = 12
y = (8 * 12) * (3/2) = 72
Therefore, the value of x is 12, and the value of y is 72.