the third term of a geometric sequence is t3= -75 and the sixth term is t6= 9375. determine the first term and the common ratio

For a geometric series,

the nth term is
t(n)=arn

So
t(3)=ar³
t(6)=ar6
t(6)/t(3)
= ar6 / ar³
= r6-3
= r³

Solve for r in
r³=9375/(-75)
r=(-125)1/3
=-5

A = t(3)/r³
= -75/(-5)³
= 3/5

and first term
= t(1)
= Ar
= 3/5(-5)
= -3

To determine the first term and the common ratio of a geometric sequence, we can use the formula for the nth term of a geometric sequence:

t_n = a * r^(n-1),

where t_n is the nth term, a is the first term, r is the common ratio, and n is the position of the term.

We are given two pieces of information: t_3 = -75 and t_6 = 9375. Let's use these values to find the first term (a) and the common ratio (r).

Step 1: Find the common ratio (r):

We can use the formula t_n = a * r^(n-1) to set up two equations with the given information:

t_3 = a * r^(3-1), (equation 1)
t_6 = a * r^(6-1). (equation 2)

Substituting the given values, we have:

-75 = a * r^2,
9375 = a * r^5.

Step 2: Solve for r:

Now, let's divide the equation 2 by equation 1 to eliminate the first term (a):

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

Simplifying the equation gives us:

-125 = r^3.

Taking the cube root of both sides, we find:

r = -5.

Step 3: Find the first term (a):

Substituting the value of r (-5) into either equation 1 or 2, we find:

-75 = a * (-5)^2,
-75 = a * 25.

Simplifying the equation gives us:

a = -3.

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

To determine the first term and the common ratio of a geometric sequence, you can use the following formula:

tn = a * r^(n-1)

Where:
tn represents the nth term of the sequence,
a represents the first term of the sequence,
r represents the common ratio of the sequence,
n represents the position of the term in the sequence.

We are given the values of t3 and t6, so we can substitute those into the formula:

t3 = a * r^(3-1)
-75 = a * r^2

t6 = a * r^(6-1)
9375 = a * r^5

Now we can solve the equations simultaneously to find the values of a and r.

Dividing the second equation by the first equation, we get:

9375 / -75 = (a * r^5) / (a * r^2)
-125 = r^3

Taking the cube root of both sides, we have:

r = -5

Substituting this value back into the second equation, we find:

9375 = a * (-5)^5
9375 = a * (-3125)
a = -3

Therefore, the first term (a) in the geometric sequence is -3, and the common ratio (r) is -5.