if 4,2x-4,......,64,21x+2 are the terms of GP . find the value of x ?
if GP , then
(2x-4)/4 = (21x+2)/64
128x-256 = 84x + 8
44x = 264
x = 6
check:
4, 8, ... , 64, 128
YUP, common ratio of 2
To find the value of x in the given geometric progression (GP), we need to determine the common ratio (r) between consecutive terms.
We can use the formula for the nth term of a GP: 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.
Given the terms:
4, 2x - 4, ..., 64, 21x + 2
The first term (a1) is 4, and the common ratio (r) can be found by dividing each term by its preceding term:
(2x - 4) / 4 = (21x + 2) / 64
Simplifying this equation, we get:
(2x - 4) / 4 = (21x + 2) / 64
Multiplying both sides by 64 and then simplifying further:
16x - 32 = 84x + 8
Combining like terms:
-68x = 40
Dividing both sides by -68:
x = -40 / 68
Simplifying this fraction, we get:
x = -10 / 17
Therefore, the value of x in the geometric progression is -10/17.
To find the value of x in a geometric progression (GP), we can use the property that each term in a GP is obtained by multiplying the previous term by a common ratio.
Given the terms: 4, 2x-4, ..., 64, 21x+2
Let's calculate the common ratio:
We know that the common ratio (r) can be found by dividing any term in the GP by its previous term.
For example, we can take the second term (2x-4) and divide it by the first term (4):
(2x-4)/4 = (x-2)/2
Now, let's take a look at the fifth term (64) and the fourth term:
64 / (2x-4)
Since these terms are part of the same GP, the common ratio should be the same. Therefore, we can set up an equation using the two expressions for the common ratio:
(x-2)/2 = 64 / (2x-4)
To solve this equation and find the value of x, we can cross-multiply:
(2x-4)(x-2) = 64 * 2
Now, expand and simplify:
2x^2 - 4x - 4x + 8 = 128
Combine like terms:
2x^2 - 8x + 8 = 128
Subtract 128 from both sides:
2x^2 - 8x - 120 = 0
Now, let's factor the quadratic equation:
2(x^2 - 4x - 60) = 0
2(x + 6)(x - 10) = 0
From the factored form, we see that x can be either -6 or 10.