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Exercises Sequences, counting and probability

Example 4 from Algebra and Trigonometry, 13.4 Series and Their Notations

Finding the First n Terms of a Geometric Series

Use the formula to find the indicated partial sum of each geometric series.

ⓐ $S_{11}$ for the series $8 + -4 + 2 + \dots$
ⓑ ${\sum^{}}_{k = 1}^{6} 3 \cdot 2^{k}$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ
  $a_{1} = 8,$ and we are given that $n = 11.$
  We can find $r$ by dividing the second term of the series by the first.
  $r = \frac{- 4}{8} = - \frac{1}{2}$
  
  Substitute values for $a_{1}, r, and n$ into the formula and simplify.
  $\begin{array}{l} S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{11} = \frac{8 (1 - {(- \frac{1}{2})}^{11})}{1 - (- \frac{1}{2})} \approx 5.336 \end{array}$
2. ⓑ
  Find $a_{1}$ by substituting $k = 1$ into the given explicit formula.
  $a_{1} = 3 \cdot 2^{1} = 6$
  We can see from the given explicit formula that $r = 2.$ The upper limit of summation is 6, so $n = 6.$
  Substitute values for $a_{1}, r,$ and $n$ into the formula, and simplify.
  $\begin{array}{l} S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{6} = \frac{6 (1 - 2^{6})}{1 - 2} = 378 \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

$S_{20}$ for the series $1,000 + 500 + 250 + \dots$

How did it go?

PreviousSolving Application Problems with Arithmetic Series Next Solving an Application Problem with a Geometric Series

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 13.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.