GEOMETRIC SERIESMr. Jhon Paul A. Lagumbay

Mathematics Teacher

Definition:

A geometric series is the sum of the terms of a geometric sequence. If the number of terms in the geometric sequence is finite, the sum of terms is called a finite geometric series. If not, the series is infinite.

The sum of the first n terms in a geometric sequence is given by

where

= the sum of the first n terms

= the first term,

= the common ratio, and

= the number of terms.

Problem 1

1. Find the sum of the first seven terms of the geometric sequence -5, 10, -20, 40, …

2. Find the sum of the first six terms of the geometric sequence 2, -8, 32, -128, …

Problem 2

Angelo and his employer agreed that he will be paid based on the following scheme: 1 peso on the first day, 2 pesos on the second day, 4 pesos on the third day, 8 pesos on the fourth day, and so on. How much would Angelo receive after working for 15 days with no absences?

Problem 3

A bacteria culture started out with 500 bacteria and doubles in number every hour. How many bacteria will be reproduced after 12 hours?

Problem 4

Mike deposited Php 850 into the bank in July. From July to December, the amount of money which he deposited into the bank increased by 25% per month. What is the total amount of money in his account after December?

Problem 5

A ball tossed to a height of 4 meters rebounds to 40% of its previous height. Find the distance the ball has traveled when it strikes the ground for the fifth time.

Problem 6

A contract specifies that Marlyn Figueroa will receive a 5% pay increase each year for the next 30 years. She is paid Php 120,000.00 the first year. What is her total lifetime salary over a 30-year period?

Review Problem

Suppose you save 1 peso on the first day of a month, 2 pesos on the second day, 4 pesos on the third day, and so on. That is, each day you save twice as much as you did the day before.a. What will you put aside for saving on

the 15th day of the month?b. How much will you saved in all on the

30th day of the month?

EXPLORE!!!PARADOX OF ZERO

Here’s a paradox devised by Zeno 2500 years ago:

To go from Athens to Sparta, first you must travel half the distance. Then, you must again travel half of the remaining distance each time. Will you ever reach Sparta?

Athens Sparta1st 2nd 3rd 4th 5th

𝟏𝟐

𝟏𝟒

𝟏𝟖

𝟏𝟏𝟔

𝟏𝟑𝟐

𝟏𝟔𝟒

6th

SEQUENCE:

Infinite Geometric SeriesIf , then the infinite geometric series

converges into a particular value.

Then, the sum of an infinite geometric series is given by:

The series converges because each term gets smaller and smaller (since ).

Problem 7

1. Find the sum of the infinite geometric series

2. Find the sum of this infinite geometric series

Problem 8Determine whether each geometric series

converges. If so, find the sum of each geometric series.

1.

2.

3.

4.

5.

6.

Problem 9

1. Show that the repeating decimals equals .

2. Show that the repeating decimals equals .

Problem 10

A ball tossed to a height of 4 meters rebounds to 40% of its previous height. Find the total distance travelled by the ball by the time it comes to rest.