Section 1.4 Geometric Series

PRE CALCULUS 11SECTION 1.4GEOMETRIC SERIES

i) Sums of a geometric series and infinitegeometric series

ii) Deriving the formula for the sum of ageometric series and infinite geometric series

DEFINITION: GEOMETRIC SERIES

A geometric series is the sum [addition] of theterms in a geometric sequence

Consider the geometric sequence 1, 2, 4, 8, 16,32, 64, 128, 256, 512, 1024 .

If we add the terms of the sequence, we canwrite the geometric series as

 In this section we will learn to find the value ofa geometric series

II) SUM OF GEOMETRIC SEQUENCE:

Sum of a geometric series up to the nth term

Multiply both sides by “r”

Subtract the equations!

Factor out “Sn”

Divide both sides by (1 - r)

Factor out “a”

ALTERNATIVE EQUATION:

Since the last term tn is:

We can rewrite the equation as:

The sum of ageometric seriesfrom the first term tothe nth term

Firstterm (a)

Common ratio timesthe last term

One minus the common ratio

Sum of the first 2 terms

Sum of the first 5 terms

EX: FIND THE SUM OF THE FOLLOWING GEOMETRICSEQUENCE:

PRACTICE: DETERMINE THE SUM OF 14 TERMS OF THEGEOMETRIC SERIES: S = 6 + 18 + 54 + …

II) FIND THE SUM OF THE FIRST 20 TERMS.

EXAMPLE 4:DETERMINE THE NTH TERM, AND THE SUM OF THE FIRST NTERMS OF THE GEOMETRIC SEQUENCE WHICH HAS 2, 6, AND 18AS ITS FIRST THREE NUMBERS.

The general term is an equation in terms of “n”. Thevalue of the Geometric series varies as the value of “n”changes

EX: FIND THE SUM:

First find outhow many terms there are

Then find thesum up to the “14th”term

Since “n” is a wholenumber we can guess andcheck

Jason gave his son a penny on the first day of the monthand doubled the amount each day. How much moneywill he give his son altogether by the 30th day?

EXAMPLE 5:THE FOLLOWING IS A SCHOOL TRIP TELEPHONING TREE.

Level 1: Teacher

Level 2: Students

Level 3: Students

a) At what level are 64 students contacted?

r = 2; (Each student contacts 2 students.)

Find n .

B) HOW MANY ARE CONTACTED AT THE 8TH LEVEL?

Remember that the teacher is not counted. So we will subtract 1

from the total S8 .

The total number of students

contacted is 255 – 1 = 254 .

C) By the 8th level, how many students, in total, have been contacted?

D) BY THE NTH LEVEL, HOW MANY STUDENTS, IN TOTAL, HAVEBEEN CONTACTED?

Again, remember that the teacher is not counted.

So we will simply subtract 1 from the sum up to level n.

The total number of students by level n is given by

e)If there are 300 students in total, by what level will all have been

contacted?

Since 254 students have been contacted by level 8,

then all 300 will have been telephoned by level 9.

EX: A RUBBER BALL IS DROPPED FROM A HEIGHT OF 10M. AFTEREACH BOUNCE, THE BALL RETURNS TO 65% OF ITS PREVIOUSHEIGHT. CALCULATE ALL THE VERTICAL DISPLACEMENT RIGHTBEFORE THE 8TH BOUNCE.