Section 4
Geometric Series
Name Period #
Suppose one evening, you text two people within 5 minutes to spread some gossip. Those two people then both text two new people within 5 minutes to pass along the same gossip, and so on. The number of people texted in each 5 minute period forms a geometric sequence.
TIME |8:00 |8:05 |8:10 |8:15 |8:20 |8:25 |8:30 | |People Contacted |1 |2 |4 |8 |16 |32 |64 | |
The TOTAL number of people that know the rumor now at each time is a SUM of the terms of the geometric sequence.
Definition 1: The sum of the terms of a geometric sequence forms a geometric _______________________.
[pic] 1 person knows the rumor at 8:00 PM (YOU)
[pic] 3 people know the rumor at 8:05 PM
[pic] 7 people know the rumor at 8:10 PM
AND SO ON…
THE FORMULA FOR EVALUATING PARTIAL SUMS:
Ex 1: Using sigma notation to find partial sums of geometric series looks very familiar to what we did with arithmetic series:
Give the sum [pic]
This is equal to 2(3) + 2(32) + 2(33) + …+ 2(312)
So use the formula to give the sum [pic] =
2) Eric calls 4 people within 10 minutes of his son’s birth to pass along the news. They each pass along the news to 4 new people within 10 minutes, and so on. How many total people have heard the news in 1 hour?
3) Write the geometric series 3 + 6 + 12 + … + 96 in sigma notation and find its sum.
3) Write the geometric series 1 + 4 + 16 + … + 4096 in sigma notation and find its sum.
4) Give the sum [pic]
Geometric Series Practice Worksheet 1
Name Period #
Examine the geometric sequence 2, 6, 18, … for questions 1 – 3.
1) Give the common ratio ______________
2) Give the fourth and fifth term ______________
3) Give the partial sums S1, S2, S3, S4
S1 = __________________, S2 = __________________, S3 = __________________, S4 = __________________
Write the word or phrase that correctly completes each statement below.
4) A geometric __________ is the sum of the terms in a geometric ________________.
5) Add the first five terms of a geometric series to find the fifth partial ____________.
Give each sum.
6) the first 15 terms of 2 + 6 + 18 + 54 + … ______________
7) the first 10 terms of 100 + 20 + 4 + … ______________
8) 3 + 6 + 12 + … + 6,144 ______________
Write each geometric series in SIGMA NOTATION and find its SUM.
9) 3 + 12 + 48 + … + 12,288 10) 200 + 100 + 50 + … + 1.5625
Sigma Notation: ___________ Sigma Notation: ___________
Sum: ____________________ Sum: ____________________
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The partial sum of the first n terms of a geometric series with first term a1 and common ratio [pic] is given by the formula:
[pic]
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