Find the rule for each geometric sequence
a3 = -4, a6 = 500
well, a6 = a3 * r^3, so r^3 = -500/4 = -125
so, r = -5
now you can find a, and of course, as for all GP's
an = arn-1
To find the rule for a geometric sequence, we need to identify the common ratio (r).
Given that a3 = -4 and a6 = 500, we can use these values to determine the ratio between consecutive terms.
The formula for the nth term of a geometric sequence is given by:
an = a1 * r^(n-1)
Let's find the common ratio (r) using the given information.
First, we have:
a6 = a1 * r^(6-1) = 500
And also:
a3 = a1 * r^(3-1) = -4
Now, we can solve these two equations to find the common ratio:
-4 = a1 * r^2 ---(1)
500 = a1 * r^5 ---(2)
Dividing equation (2) by equation (1) gives us:
500 / -4 = (a1 * r^5) / (a1 * r^2)
Simplifying, we have:
-125 = r^3
Taking the cube root of both sides, we find:
r = -5
So, the common ratio (r) for this geometric sequence is -5.
Therefore, the rule for the geometric sequence can be written as:
an = a1 * (-5)^(n-1)
To find the rule for a geometric sequence, we need to determine the common ratio (r) of the sequence.
Given the terms:
a3 = -4
a6 = 500
We can use these two terms to find the common ratio by dividing a6 by a3:
r = a6 / a3
Substituting the given values:
r = 500 / -4
Simplifying:
r = -125
Now that we have the common ratio, we can find the first term (a1) by using the formula:
a1 = a3 / r^2
Substituting the given value for a3 and the calculated value for r:
a1 = -4 / (-125)^2
Simplifying:
a1 ≈ -4 / 15625
a1 ≈ -0.000256
Therefore, the rule for the geometric sequence is:
an = a1 * r^(n-1)
Plugging in the values:
an = -0.000256 * (-125)^(n-1)
So, the rule for the geometric sequence is:
an = -0.000256 * (-125)^(n-1)