Find a1 and r for the geometric sequence: a4=

-1/4 and a9= -1/28

To find the values of a1 and r for a geometric sequence, we can use the formulas:

an = a1 * r^(n-1)

where an represents the nth term of the sequence, a1 is the first term, r is the common ratio, and n is the term number.

Given that a4 = -1/4 and a9 = -1/28, we can use these values to set up two equations:

a4 = a1 * r^(4-1)
-1/4 = a1 * r^3 -- (Equation 1)

a9 = a1 * r^(9-1)
-1/28 = a1 * r^8 -- (Equation 2)

Now, we have a system of two equations with two unknowns (a1 and r). To solve this system, we can divide Equation 2 by Equation 1:

-1/4 / -1/28 = (a1 * r^3) / (a1 * r^8)

28/4 = r^8 / r^3
7 = r^5

Taking the fifth root of both sides, we find:

r = ∛7

Now that we have the value of r, we can substitute it back into Equation 1 to solve for a1:

-1/4 = a1 * (∛7)^3
-1/4 = a1 * 7^(3/2)
-1/4 = a1 * 7(√7)

To simplify this equation, we need to rationalize the denominator:

-1/4 = a1 * (7√7) / (√7 * √7)
-1/4 = a1 * (7√7) / 7
-1/4 = a1 * √7 / 1
-1/4 = a1 * √7

Thus, a1 = -1/4√7.

Therefore, the values of a1 and r for the given geometric sequence are:
a1 = -1/4√7
r = ∛7

to find r, note that

a9/a4 = r^5

then use that to find a1