<< Chapter < Page Chapter >> Page >
( 1 2 ) 10 = 1 1 , 024 ( 1 2 ) 20 = 1 1 , 048 , 576 ( 1 2 ) 30 = 1 1 , 073 , 741 , 824

As n gets very large, r n gets very small. We say that, as n increases without bound, r n approaches 0. As r n approaches 0, 1 r n approaches 1. When this happens, the numerator approaches a 1 . This give us a formula for the sum of an infinite geometric series.

Formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series with −1 < r < 1 is

S = a 1 1 r

Given an infinite geometric series, find its sum.

  1. Identify a 1 and r .
  2. Confirm that 1 < r < 1.
  3. Substitute values for a 1 and r into the formula, S = a 1 1 r .
  4. Simplify to find S .

Finding the sum of an infinite geometric series

Find the sum, if it exists, for the following:

  1. 10 + 9 + 8 + 7 +
  2. 248.6 + 99.44 + 39.776 +  
  3. k = 1 4 , 374 ( 1 3 ) k 1
  4. k = 1 1 9 ( 4 3 ) k
  1. There is not a constant ratio; the series is not geometric.
  2. There is a constant ratio; the series is geometric. a 1 = 248.6 and r = 99.44 248.6 = 0.4 , so the sum exists. Substitute a 1 = 248.6 and r = 0.4 into the formula and simplify to find the sum:

    S = a 1 1 r S = 248.6 1 0.4 = 414. 3 ¯
  3. The formula is exponential, so the series is geometric with r = 1 3 . Find a 1 by substituting k = 1 into the given explicit formula:

    a 1 = 4 , 374 ( 1 3 ) 1 1 = 4 , 374

    Substitute a 1 = 4 , 374 and r = 1 3 into the formula, and simplify to find the sum:

    S = a 1 1 r S = 4 , 374 1 ( 1 3 ) = 3 , 280.5
  4. The formula is exponential, so the series is geometric, but r > 1. The sum does not exist.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding an equivalent fraction for a repeating decimal

Find an equivalent fraction for the repeating decimal 0. 3 ¯

We notice the repeating decimal 0. 3 ¯ = 0.333... so we can rewrite the repeating decimal as a sum of terms.

0. 3 ¯ = 0.3 + 0.03 + 0.003 + ...

Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.

...

Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have

S n = a 1 1 r = 0.3 1 0.1 = 0.3 0.9 = 1 3 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the sum, if it exists.

k = 1 0.76 k + 1

The series is not geometric.

Got questions? Get instant answers now!

k = 1 ( 3 8 ) k

3 11

Got questions? Get instant answers now!

Solving annuity problems

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity    is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest , compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.

We can find the value of the annuity right after the last deposit by using a geometric series with a 1 = 50 and r = 100.5 % = 1.005. After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask