Determine if the sequence is geometric. If it is, find the common ratio, the 8th term, and the explicit formula. 3, -18, 108, -648,

a.) Yes it is a geometric sequence and common ratio = -6

b.) 8th term = -839808

c.) explicit formula = 3×(-6^n-1)

Step-by-step explanation:

Geometric sequences are defined by the formula ar^n-1

Where a is the first term,

r is the common ratio

n is the number of term.

a.) To check if the given series is a geometric progression, we divide 3rd term by 2nd term, and divide 4th term by 3rd them. If we get the same answer, then it is a geometric progression and this answer is the common ratio

The terms are 3, -8, 108, -648

3rd term/2nd term = 108/-18 = -6

4th term/3rd term = -648/108 = -6.

Since both answers are equal, Then the progression is a geometric progression Where a, first term =3 and common ratio = -6

b) for 8th term, the geometric formula is denoted by: ar^8-1

=3×(-6^7) = -839808

c.) explicit formula = 3 × (6^n-1)

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1