Write the sum using summation notation, assuming the suggested pattern continues.

-9 - 3 + 3 + 9 + ... + 81

Question

Write the sum using summation notation, assuming the suggested pattern continues.

-9 - 3 + 3 + 9 + ... + 81

sleepyjess · Answer

@e.mccormick

sleepyjess · Answer

darn, clicked the wrong thing...

freckles · Answer

closed means you succeeded

freckles · Answer

lol

freckles · Answer

oh so no

freckles · Answer

so hmm we need to find a pattern

sleepyjess · Answer

I meant to click delete, but instead clicked close because it moved right as I clicked lol

freckles · Answer

so it really goes like this...-9,-3,3,9,...,81

freckles · Answer

it looks like this is an arithmetic sequence

freckles · Answer

do you see the common difference?

freckles · Answer

-3-(-9)=?
3-(-3)=?
9-3=?
all of these are equal to what?

sleepyjess · Answer

adding 6

freckles · Answer

yep do first I'm going to play with this sequence then I will go back to the series you mentioned

freckles · Answer

$$a_n=a_1+d(n-1)$$
d is the common difference 
and a_1 is the first term in the sequence

freckles · Answer

you already identified d

freckles · Answer

as 6

freckles · Answer

now a_1 is ?

sleepyjess · Answer

-9
so $a_n$=-9+6(n-1)

freckles · Answer

right we need to figure out for what number n is the last term 81

freckles · Answer

$$81=-9+6(n-1)$$
a_n is 81
but what is n
well solve this for n to find out :)

sleepyjess · Answer

81 = -9 + 6n - 6
81 = -15 + 6n
96 = 6n
16 = n

freckles · Answer

$$\text{ your answer is going \to be \in this form } \sum_{n=1}^{16}a_n$$
last step just replace a_n with what you found above

sleepyjess · Answer

The -9 + 6n?

freckles · Answer

-9+6(n-1)

sleepyjess · Answer

hmmm... I guess I will go with the option closest to that... which isn't really any of them o.O

freckles · Answer

$$\sum_{n=1}^{16}a_n=\sum_{n=1}^{16}(-9+6(n-1))$$
you could combine like terms
and there are also other ways to write the summation
but you are done 
you did what it said 
which was write as a summutation

freckles · Answer

Well let me ask you a question

freckles · Answer

do they start it at n=0 ...

freckles · Answer

or some other number

sleepyjess · Answer

I don't know what they start it at, that was all the information given :/

freckles · Answer

oh i thought they gave you choices

freckles · Answer

it sounded like you were saying oyu have to select an option

sleepyjess · Answer

oh, yes, they give choices http://prntscr.com/68pk8z

freckles · Answer

so all the choices start off with n=0

freckles · Answer

do you see that

freckles · Answer

ours starts off at n=1

freckles · Answer

we can manipulate that

freckles · Answer

$$\sum_{n=1}^{16}a_n=\sum_{n=1}^{16}(-9+6(n-1)) \  \ \sum_{n=1}^{n=16}(-9+6(n-1)) \ \sum_{n-1=0}^{n-1=15}(-9+6(n-1))$$

freckles · Answer

now replace n-1 
wherever you see n-1 with new n

freckles · Answer

$$\sum_{n=0}^{n=15}(-9+6(n))$$

sleepyjess · Answer

that makes more sense :) thanks for coming to help even though I accidentally closed it haha

freckles · Answer

do you see how i manipulated
like we had from n=1 to n=16
I subtracted 1 on both sides for these equations
n=1
n-1=1-1
n-1=0

and 
n=16
n-1=16-1
n-1=15

and our expression was already in terms of (n-1)

sleepyjess · Answer

yes

freckles · Answer

cool stuff

freckles · Answer

also the bottom two make no since because we do not have an infinite series :)
and the second one also doesn't make sense because none of the beginning terms in your sequence are multiples of -54
so I think you could have done less work here
and just used process of elimination a long with common sense
but it is also nice to see how to set it up :)

sleepyjess · Answer

I figured it was the first one, but I didn't want to just pick an option, I wanted to know how to do it

freckles · Answer

And I totally think that is cool :)