SOLUTION: Write the sum using summation notation, assuming the suggested pattern continues. 9 - 18 + 36 - 72 + ... (2 points) choices are below summation of nine times two to the powe

Algebra ->  Trigonometry-basics -> SOLUTION: Write the sum using summation notation, assuming the suggested pattern continues. 9 - 18 + 36 - 72 + ... (2 points) choices are below summation of nine times two to the powe      Log On


   

Question 1135462: Write the sum using summation notation, assuming the suggested pattern continues.
9 - 18 + 36 - 72 + ... (2 points)
choices are below
summation of nine times two to the power of n from n equals zero to infinity

summation of nine times negative two to the power of the quantity n plus one from n equals zero to infinity

summation of nine times two to the power of the quantity n plus one from n equals zero to infinity

summation of nine times negative two to the power of n from n equals zero to infinity
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


Which is the sum of 9(-2)^n from n = 0 to infinity (aka the terms go on forever)
Therefore the answer is choice D


Note how plugging n = 0 into 9(-2)^n leads to
9(-2)^n = 9(-2)^0
9(-2)^n = 9(1)
9(-2)^n = 9
So the first term to be added is 9

Then trying n = 1 leads to
9(-2)^n = 9(-2)^1
9(-2)^n = 9(-2)
9(-2)^n = -18
Making -18 to be the second term to be added. So far we have the expression

Trying n = 2 leads to
9(-2)^n = 9(-2)^2
9(-2)^n = 9(4)
9(-2)^n = 36
We add on 36 as the third term. So far we have the expression

Trying n = 3 leads to
9(-2)^n = 9(-2)^3
9(-2)^n = 9(-8)
9(-2)^n = -72
We add on -72 as the fourth term. So far we have the expression

As n heads off to infinity, we end up with the expression

side notes

  1. The sequence {9, -18, 36, -72, ...} is geometric.

  2. The starting term is a1 = 9

  3. The common ratio is r = -2

  4. The common ratio can be found by dividing any term by its previous one (eg: term3/term2 = 36/(-18) = -2)

  5. The infinite series 9-18+36-72+... diverges meaning the sum of the infinitely many terms does not approach a fixed value (ie the sum keeps drastically changing as you add on more terms). The infinite series only converges if -1 < r < 1, but r = -2 is outside this range.