Illustrates Geometric Sequence, Slides of Mathematics

Download Illustrates Geometric Sequence and more Slides Mathematics in PDF only on Docsity!

Review the nth term, arithmetic mean and the sum of arithmetic sequence; illustrate geometric sequence; differentiate geometric sequence from arithmetic sequence

Objective

s

Activity 1: REVIEW Find the NTH term, arithmetic mean and the sum of Arithmetic sequence

Determine the term which has the value -38 in the arithmetic sequence 7, 4, 1, -2, … Given:

𝒏

𝟏

𝒏 =¿ 𝒅 =¿

− 𝟑𝟖 = 𝟕 − 𝟑 𝒏 + 𝟑 − 𝟑𝟖 = 𝟏𝟎 − 𝟑 𝐧 − 𝟑𝟖 − 𝟏𝟎 = − 𝟑 𝒏 Solution:

𝒏

𝟏

Formula:

- ? 𝒏 = 𝟏𝟔 𝟕 − 𝟒𝟖 = − 𝟑 𝒏 -3 -

Find if Given:

𝟗

𝟏

𝒏 =¿ 𝒅 =¿

𝒏

𝟏

𝟏 Solution:

𝒏

𝟏

Formula: 𝟔

𝟏

Insert three arithmetic means between 6 and 54. Given:

𝒏

𝟏

𝒏 =¿

𝒏

𝟏

𝟓𝟒 = 𝟔 +( 𝟓 − 𝟏 ) 𝒅 𝟓𝟒 = 𝟔 + 𝟒 𝒅 𝟓𝟒 − 𝟔 = 𝟒 𝒅 Solution:

𝒏

𝟏

Formula: ? 5 𝒅 = 𝟏𝟐

6, ___,^ ___,^ ___,^54

𝟒𝟖 = 𝟒 𝒅 -4 -

Find the sum of the first 20 positive even numbers Solution: Formula: 2, 4, 6, 8,10, …

𝟏

𝒏 =¿ 𝒅 =¿ 𝟐 𝟐𝟎 𝟐 𝑺 𝒏 = 𝒏 𝟐 [ 𝟐 𝑨 𝟏 +(^ 𝒏 − 𝟏 )^ 𝒅 ] 𝑺 𝒏 = 𝟐𝟎 𝟐 [ 𝟐 ( 𝟐 )+(^ 𝟐𝟎 − 𝟏 )^ 𝟐 ] 𝑺 𝒏

= 𝟏𝟎 [ 𝟒 +( 𝟏𝟗 ) 𝟐 ]

𝒏

= 𝟏𝟎 [ 𝟒 + 𝟑𝟖 ]

𝒏

= 𝟏𝟎 [ 𝟒𝟐 ]

𝒏

Geometric Sequence

Activity 1: Fold Me Up

1. Get a paper. Fold the paper in half once. How many rectangles are formed? Record your answer in the table for each trial.
2. Using the same sheet of paper, repeat the process 3 times. Number of Folds 0 1 2 3 4 Number or Rectangles

The common ratio ( r) can be determined by dividing any term in the sequence by the term that precedes it. 𝒓 = 𝑨 𝟐 𝑨 𝟏 = 𝑨 𝟑 𝑨 𝟐 = 𝑨 𝟒 𝑨 𝟑 = 𝑨 𝟓 𝑨 𝟒

𝟏

𝟐

𝟑

𝟒

𝟓

1, 2, 4, 8, 16, … 𝒓^ =

Example: (^2 2 2 )

Identify the common ratio and the next term of the following geometric sequences.

sequence

r

next term

(^4 )

. -27, 9, -3, 1, … (^) - 8 1/ 3 - 1/ 3 1, 5, 25, 125, … 5 62 5 . -4, -2, -1, -1/2, … 1/ 2

2, -8, 32, -128, … (^) - 1/ 51

Differentiate arithmetic sequence from a geometric sequences. Arithmetic Sequence Geometric Sequence is a sequence where every term after the first is obtained by ADDING a CONSTANT called the COMMON DIFFERENCE. is a sequence where each term after the first is obtained by MULTIPLYING the preceding term by a FIXED NUMBER called a COMMON RATIO.

Arithmetic Sequence Geometric Sequence Differentiate arithmetic sequence from a geometric sequences. The common difference (d) can be determined by SUBTRACTING any term in the sequence by the terms that precedes it. The common ratio (r) can be determined by DIVIDING any term in the sequence by the term that precedes it.

Tell whether the sequence is an

Arithmetic Sequence or Geometric

Sequence

1. 3, 5, 7, 9, …
2. 4, 12, 36, 108, …
3. 3, -3, -9, -15, …
4. 36, 29, 22, 15, Arithmetic Geometric Arithmetic Arithmetic Geometric

d = 2

r = 3

d = -

d = -

r = -

Geometric r = 5

Check Your Understanding What have you learned from today’s lesson?

1. _____________ is a constant multiplied to each term of a geometric sequence to obtain the next term of the sequence.
   2. In the sequence -2, -10, -50, ... the common ratio is _______.
   3. A geometric sequence, or geometric progression, is a sequence where each term after the first is obtained by

   ---

   __________________________________. 

Common ratio

preceding term by a fixed number multiplying the

r