How do you use sigma notation to write the sum for #3-9+27-81+243-729#?

Answer 1

#sum_(n=0)^5 [3*("–"3)^n]" "# or #" "-sum_(n=1)^6 ("–"3)^n" "# or #" "sum_(n=1)^6 ("–"1)^(n-1)3^n#

In order to use summation notation, we need to find something that increments by 1 with each new term.

What we need to do is discern how each successive term differs from the previous one. Without looking too hard, we can see that each term is #"–"3# times the previous term. Thus, starting at our first term of 3, each new term is "3 times the next power of #"–"3#":
#"    "3=3# #"  –"9=3xx("-"3)^1# #"  "27=3xx("-"3)^2# #"–"81=3xx("-"3)^3#

and so on.

What we notice is that the powers of -3 are going up by 1 with each new term. This is the increment value we will use in our sigma notation.

Our sum can be thought of as:

#3-9+27-81+243-729# #=3+("-"9)+27+("-"81)+243+("-"729)# #=[3*("-"3)^0]+[3*("-"3)^1]+[3*("-"3)^2]+[3*("-"3)^3]+[3*("-"3)^4]+[3*("-"3)^5]#

Which is written in sigma notation as:

#sum_(n=0)^5 [3*("–"3)^n]#
This says we wish to sum the expression #3*("–"3)^n# a total of 6 times—once for each #n# from #{0,1,2,3,4,5}#.

There are other ways to write the sum; they are included as extras in the Answer portion.

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Answer 2

The sum can be represented using sigma notation as follows:

∑((-3)^n), where n starts from 0 and ends at 5.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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