Question

The sum of first three terms of a finite geometric series is -7/10 and their product is -1/125. [Hint: Use , a, and ar to represent the first three terms, respectively.] The three numbers are _____, _____, and _____.

Answer 1

First, you know what a geometric sequence is? (It says series but they're closely related).

Answer 2

yeah, i know it's finite so it has an end.

Answer 3

Yup. The sequence is just three terms that have a common ratio, call it r, with the first term, which we can call a.
So the sequence, which is what the hint is providing, looks like this:
a, a*r, a*r^2 (first three terms)
Then we can set up two equations using the two pieces of information:
the SUM: add our three terms, and we get: = -7/10.
the PRODUCT, we multiply the three terms, and we get = -1/125.
Are you able to set up these two equations from that information?

Answer 4

well i'm having trouble with figuring out how i would find that out. would i just pick anything that adds up to that?
i kind of understand it a little better

Answer 5

let me think for a second

Answer 6

Basically, we are going to get two equations, and we'll have two variables in each -- a and r.
We can always solve a system of equations like this just by using one equation to solve for a variable, then plugging it into the other to solve for one variable's answer.

Answer 7

would it just be whatever adds up to -7/10 that also when multiplied equals -1/125?

Answer 8

You would have the three terms that add up to -7/10:
a + ar + ar^2 = -7/10
and similarly for the product for -1/125, using the same a, ar, and ar^2.

Since our three terms specifically also follow the rule of a geometric sequence, so that is also something we account for by using a, ar, and ar².

Answer 9

If you just found three numbers that added up to -7/10 and multiplied to -1/125, you would just have three arbitrary variables:
x + y + z = -7/10
x*y*z = -1/125 But now we have three variables for two equations. The geometric series information gives us information to eliminate a variable because otherwise, this cannot be solved properly for actual solutions -- we'll get a dummy variable.

Answer 10

what does it mean that's it ar instead of just one?

Answer 11

that it's*

Answer 12

You know that the geometric sequence has this form?
a first term
a*r second term
a*r^2 third term
We're adding up those first three terms? a + ar + ar^2 = -7/10

Answer 13

i'm not that familiar with it because my teacher just started going into it yesterday. but it's just any numbers then? i'm having trouble trying to figure it out how to get -7/10

Answer 14

mostly with something that has an exponent

Answer 15

Guessing the answer will not get us far because this is a relatively complicated equation.
Have you dealt with quadratic equations in the past?

Answer 16

i'm not sure, i don't really remember what it is. i think so, but if i did it was a while ago

Answer 17

wait no, i just looked it up and i've never done that.

Answer 18

No experience with quadratic equations? Like,
\( x^2 + ax + b = 0 \) ?
That might make this quite difficult.

Answer 19

oh wait no, i've done that.

Answer 20

let the terms be \[\frac{ a }{ r},a,ar\]
then\[\frac{ a }{ r }\times a \times ar=-\frac{ 1 }{ 125}\]
\[a^3=-\frac{ 1 }{ 125 }=\left( -\frac{ 1 }{ 5 } \right)^3,a=\frac{ -1 }{ 5 }\]
terms are \[-\frac{ 1 }{ 5r },-\frac{ 1 }{ 5 },-\frac{ r }{ 5 }\]
again\[-\frac{ 1 }{ 5r }-\frac{ 1 }{ 5 }-\frac{ r }{ 5 }=-\frac{ 7 }{ 10 }\]
multiply each term by-10r and simplify to form a quadratic and find the values of r.
Then three terms.

Answer 21

Oh I see, I knew I should have asked for clarification when I saw the blank where the first term was. I hadn't thought about using a/r, a, and ar for the three terms! :D

Answer 22

okay i think i get it now, thanks both of you for your help.

Answer 23

yw

Answer 24

oh i thought i could choose more than one best response :( sorry about that

Answer 25

It's no problem! I learned something today so all is well. Glad to help! :P