The third and sixth terms of a geometric sequence are -75 and -9375 respectively. Find the first term, the common ratio, and an explicit rule for the nth term.

t3 = ar^2 = -75

t6 = ar^5 = -9375
divide them
r^3 = 125
r = 5

in ar^2 = -75
a(25) = -75
a = -3

term(n) = -3(5)^(n-1)

To find the first term, common ratio, and explicit rule for the nth term of a geometric sequence, we can use the given information.

Let's denote the first term as 'a' and the common ratio as 'r'.

We are given the third term as -75, so we can express this as:

a * r^2 = -75 ...(1)

Similarly, we are given the sixth term as -9375, which can be expressed as:

a * r^5 = -9375 ...(2)

Now, we can solve these two equations simultaneously to find the values of 'a' and 'r'.

Dividing equation (2) by equation (1), we get:

(a * r^5) / (a * r^2) = -9375 / -75

Simplifying, we have:

r^3 = 125

Taking the cube root of both sides, we get:

r = 5

Substituting this value of 'r' into equation (1), we can find 'a':

a * (5^2) = -75

25a = -75

Dividing both sides by 25, we obtain:

a = -3

Therefore, the first term of the geometric sequence is -3 and the common ratio is 5.

To find the explicit rule for the nth term of the sequence, we can use the formula:

an = a * r^(n-1)

Substituting the values we found, the explicit rule for the nth term is:

an = -3 * 5^(n-1)

To find the first term, common ratio, and explicit rule for the nth term of a geometric sequence, we can use the given information:

Let's call the first term of the sequence "a" and the common ratio "r".

Given that the third term is -75, we know that the formula for the third term of a geometric sequence is a * r^2 = -75. (Since the power of r is 2 for the third term).

Similarly, given that the sixth term is -9375, we have a * r^5 = -9375.

Now, we have two equations and two variables to solve for. Let's solve it step by step:

Equation 1: a * r^2 = -75

Equation 2: a * r^5 = -9375

Dividing Equation 2 by Equation 1, we get:

(a * r^5) / (a * r^2) = -9375 / -75

Simplifying, we have:

r^3 = 125

We can rewrite 125 as 5^3:

r^3 = (5^1)^3

Using the rule of exponents, r^3 = 5^(1*3) = 5^3

Therefore, r = 5.

Now, substitute the value of r = 5 into Equation 1 to find the value of "a":

a * (5)^2 = -75

25a = -75

Dividing both sides by 25, we get:

a = -3

So, the first term (a) is -3, and the common ratio (r) is 5.

To find an explicit rule for the nth term, we can use the general formula for a geometric sequence:

nth term = a * r^(n-1)

Substituting the values of a = -3, r = 5, and simplifying, we get:

nth term = -3 * 5^(n-1)

Therefore, the explicit rule for the nth term of the geometric sequence is -3 * 5^(n-1).