in a geometric sequence,the fifth term is four times the third term,and the second term is 4.if r<0 determine the value of r,the common ratio
a r^4 = 4 a r^2
r^2 = 4
r = +/- 2
-2
To find the value of the common ratio (r) in a geometric sequence, we can use the given information.
Let's denote the first term as a₁, the second term as a₂, the third term as a₃, and so on.
We are given that the second term is equal to 4, so a₂ = 4.
We are also told that the fifth term (a₅) is four times the third term (a₃), which can be written as a₅ = 4a₃.
In a geometric sequence, each term after the first is found by multiplying the previous term by the common ratio (r). So we have the following relationships:
a₂ = a₁ * r,
a₃ = a₂ * r = a₁ * r * r,
a₅ = a₄ * r = (a₁ * r * r * r) * r.
Now, let's substitute the known values into these equations:
4 = a₁ * r,
4a₃ = a₁ * r * r,
4a₃ = (a₁ * r * r) * r.
Since a₂ = 4, we can rewrite the first equation as a₁ * r = 4.
Substituting a₁ * r = 4 into the second equation, we get:
4a₃ = 4 * r,
a₃ = r.
Similarly, substituting a₁ * r = 4 into the third equation, we get:
4a₃ = 4 * r * r * r,
a₃ = r * r * r.
Now, we can equate the two expressions for a₃:
r = r * r * r.
To solve this equation, we can divide both sides by r:
1 = r * r * r / r,
1 = r * r²,
1 = r³.
Now, we have a cubic equation: r³ = 1.
Since we are given that r < 0, we need to find the negative cube root of 1, which is -1.
Therefore, the value of r (the common ratio) when r < 0 is -1.