Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the sum of the infinite geometric series.\newline−10−52−58−532+-10 - \frac{5}{2} - \frac{5}{8} - \frac{5}{32} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline______

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Precalculusright chevron icon
Find the value of an infinite geometric seriesright chevron icon

Full solution

Q. Find the sum of the infinite geometric series.\newline−10−52−58−532+-10 - \frac{5}{2} - \frac{5}{8} - \frac{5}{32} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline______
  1. Identify Terms: First, we need to identify the first term aa and the common ratio rr of the geometric series.\newlineThe first term aa is −10-10.\newlineTo find the common ratio rr, we divide the second term by the first term: r=−5/2−10=14r = \frac{-5/2}{-10} = \frac{1}{4}.
  2. Calculate Common Ratio: Now that we have the first term and the common ratio, we can use the formula for the sum of an infinite geometric series, which is S=a(1−r)S = \frac{a}{(1 - r)}, provided that ∣r∣<1|r| < 1. In this case, ∣14∣<1|\frac{1}{4}| < 1, so we can use the formula.
  3. Use Sum Formula: Let's plug the values into the formula:\newlineS=−101−14S = \frac{-10}{1 - \frac{1}{4}}\newlineS=−1034S = \frac{-10}{\frac{3}{4}}\newlineS=−10×43S = -10 \times \frac{4}{3}
  4. Plug in Values: Now, we perform the multiplication to find the sum:\newlineS=−403S = -\frac{40}{3}\newlineThis is the sum of the infinite geometric series in fraction form.

More problems from Find the value of an infinite geometric series

Question
Find the sum of the finite arithmetic series. ∑n=110(7n+4)\sum_{n=1}^{10} (7n+4)\newline______
Get tutor helpright-arrow

Posted 2 months ago

Question
Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`
Get tutor helpright-arrow

Posted 1 month ago

Question
Type the missing number in this sequence: `2, 4 8`, ______,`32`
Get tutor helpright-arrow

Posted 2 months ago

Question
What kind of sequence is this? 2,10,50,250,…2, 10, 50, 250, \ldots Choices:Choices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
Get tutor helpright-arrow

Posted 2 months ago

Question
What is the missing number in this pattern? 1,4,9,16,25,36,49,64,81,____1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_
Get tutor helpright-arrow

Posted 1 month ago

Question
Classify the series. ∑n=012(n+2)3 \sum_{n = 0}^{12} (n + 2)^3 \newlineChoices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
Get tutor helpright-arrow

Posted 1 month ago

Question
Find the first three partial sums of the series.\newline1+6+11+16+21+26+⋯1 + 6 + 11 + 16 + 21 + 26 + \cdots\newlineWrite your answers as integers or fractions in simplest form.\newlineS1=S_1 = ____\newlineS2=S_2 = ____\newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 1 month ago

Question
Find the third partial sum of the series. \newline3+9+15+21+27+33+⋯3 + 9 + 15 + 21 + 27 + 33 + \cdots\newlineWrite your answer as an integer or a fraction in simplest form. \newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 1 month ago

Question
Find the first three partial sums of the series. \newline1+7+13+19+25+31+⋯1 + 7 + 13 + 19 + 25 + 31 + \cdots\newlineWrite your answers as integers or fractions in simplest form. \newlineS1=S_1 = ____\newlineS2=S_2 = ____\newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 2 months ago

Question
Does the infinite geometric series converge or diverge?\newline1+34+916+2764+⋯1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots\newlineChoices:\newline[A]converge\text{[A]converge}\newline[B]diverge\text{[B]diverge}
Get tutor helpright-arrow

Posted 1 month ago

View all (62)