Section 2.3 Geometric Series

SECTION 2.3GEOMETRIC SERIES

i) Sums of a geometric series and infinitegeometric series

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

I) 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”

EX: FIND THE SUM OF THE FOLLOWING GEOMETRICSEQUENCE:

EX: FIND THE SUM:

First find outhow many terms there are

Then find thesum up to the “14th”term

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.

Note: There are two types ofdisplacement: Going up and down

Each displacement is doubled

except the first bounce

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?

CHALLENGE: THE SUM OF THE SECOND & THIRD TERM IN AGEOMETRIC SERIES IS 45. THE SUM OF THE FOURTH & FIFTHTERM IS 20. FIND THE GEOMETRIC SEQUENCE:

The 4th and 5th terms add to 20

The 2nd and 3rd terms add to 45

Factor out any common factors

In each equation

Divide the equations

Use the common ratio to find the first term

Find the 2nd series when thecommon ratio is negative

I) WHAT IS AN INFINITE GEOMETRIC SERIES?

A Geometric series with an infinite number of terms

If the common ratio is greater than 1, each term will getbigger and the sum will soon add up to infinity

Note: Any infinite G.S. with a common ratio then the sum willadd up to infinity

Most infinite G.S. in sect. 2.9 will have a common ratiobetween -1 and 1

Each term in the series will get smaller

Eventually some terms will become so small such thatadding them will be insignificant, like adding zero

When the ratio is the infinite G.S. will converge to afixed value

These values are so small such that adding them will be insignificant!!

II) FORMULA FOR THE SUM OF AN INFINITE G.S.

If the Ratio is bigger than 1, the sum will be positive infinity

If the ratio is smaller than -1, the sum will be negativeinfinity

If the Ratio is between 1 and -1, the sum can be obtainedthrough a formula:

Formula for the sum of a G.S. with “n” terms

If there are infinite terms, then “n” will be infinity

Since

the value of

will be zero

The equation will only apply if

EX: FIND THE SUM OF THE INFINITE G.S.

EX: THE COMMON RATIO OF AN INFINITE GEOMETRICSERIES IS 0.75. IF THE SUM OF ALL THE TERMSCONVERGES TO 20, FIND THE FIRST TERM.

GIVEN THE FOLLOWING INFINITE GEOMETRICSEQUENCE, WHAT SHOULD THE VALUE OF “R”BE SO THAT THE SUM WILL BE 20?

The series must be converging

So the value of “r” must be between -1 and 1

The answer must be r = 0.80

EX: A MOVIE IN A THEATRE GENERATE $250,000 INREVENUE IN THE FIRST MONTH OF ITS SHOWING. EACHMONTH, SALES FROM THAT MOVIE DROP BY 15%. IF THEMOVIE IS SHOWN FOR A LONG TIME, WHAT IS THE TOTALREVENUE GENERATED?

HOMEWORK:

Assignment 2.3