The first three terms of GP are x+1,(x^2-1) and (x^2-1)(2x-4).Calculate the value of x. Suggested solution;ar^2/ar,so r=2x-4 and a=x+1. Substitute in second term(ar)=(x^2-1). Factorise, so x=3
Your solution for x = 3 is correct,
now all you have to do is sub x = 3 into your 3 terms to get
4, 8, 16
you already knew r = 2 from r = 2(3) - 4 = 2
and a = 4 from a = x+1 = 4
Thanks Reiny.
To calculate the value of x in the given geometric progression (GP), we can use the concept of finding the common ratio (r) and the first term (a).
The formula for the n-th term of a GP is given by:
Tₙ = a * r^(n-1)
Here, we are given the first three terms of the GP:
Term 1: x+1
Term 2: (x^2-1)
Term 3: (x^2-1)(2x-4)
To find the common ratio (r), we can use the formula T₂/T₁ = T₃/T₂:
((x^2-1) / (x+1)) = ((x^2-1)(2x-4) / (x^2-1))
By canceling out the common factor (x^2-1), we get:
1 / (x+1) = 2x-4
Now, let's solve this equation step by step:
1. Cross-multiply:
1 = (2x-4)(x+1)
2. Expand the right side:
1 = 2x^2 + 2x - 4x - 4
3. Combine like terms:
1 = 2x^2 - 2x - 4
4. Move 1 to the other side:
2x^2 - 2x - 4 - 1 = 0
5. Simplify:
2x^2 - 2x - 5 = 0
Now, we can factorize this quadratic equation to simplify it:
(2x + 1)(x - 5) = 0
Using the zero-product property, we see that either (2x + 1) = 0 or (x - 5) = 0:
1. If (2x + 1) = 0, then x = -1/2.
2. If (x - 5) = 0, then x = 5.
So, the potential values for x are -1/2 and 5.
However, to confirm the correct value of x, let's go back to the expression for the common ratio (r):
r = 2x - 4
If we substitute x = -1/2, we get r = 2(-1/2) - 4 = -5.
If we substitute x = 5, we get r = 2(5) - 4 = 6.
Since a geometric progression cannot have a negative common ratio, we can eliminate -1/2 as the value of x.
Therefore, the value of x is 5.