ALGEBRA AND CALCULUS, Slides of Mathematics

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Learning Unit 8:

Arithmetic & Geometric Sequence

Students should be able to:

1. Identify the arithmetic and geometric sequence and series
2. Perform nth-term and sum of arithmetic and geometric series
3. Demonstrate application of arithmetic and geometric in Mathematic Finance

Learning Objectives

Sequences

f ( n ) = 2 n – 1 (where domain of f is the set of natural numbers N ) f (1) = 1, f (2) = 3, f (3) = 5, …… Can be written as: a n = 2 n – 1 a 1 = 2(1) – 1 = 1 a 2

a 3

a n = 2(n) – 1 = n th First Term Second Term Third Term General Term Term of Sequence  (^) A sequence can be thought of as a pattern of numbers listed in a prescribed order.

Series

_a 1

• a 2
• a 3_

a n 1, 2, 4, 8, 16 1 + 2 + 4 + 8 + 16 1, 2, 4, 8, 16, …. 1 + 2 + 4 + 8 + 16 + … Finite Sequence Finite Series Infinite Sequences Infinite Series

n th-Term of Arithmetic

Sequence

T

_2 = a 2, = a 1

• d =_ a + d _T 3 = a 3, = a 2
• d =_ a + 2d _T 4 = a 4, = a 3
• d =_ a + 3d T n = a + (n – 1)d ; n > 1

EXAMPLE

If the first and tenth terms of an arithmetic sequence are 3 and 30, find the fiftieth term of the sequence.

EXAMPLE

Find four arithmetic means that lie between -1 and 19.

EXAMPLE 2 SOLUTION Find four arithmetic means that lie between –1 and 19. –1, p , q , r , s , 19; a = –1, T 6

T

n = a + ( n –1) d T 6 = a + (6 – 1) d 19 = –1 +5 d 20 = 5 d d = 4 p = –1 + (2 – 1)4 = 3 q = 3 + 4 = 7 r = 7 + 4 = 11 s = 11 + 4 = 15

EXAMPLE

Find the sum of the first 26 terms of an arithmetic series if the first term is -7 and d = 3.

EXAMPLE 3 SOLUTION Find the sum of the first 26 terms of an arithmetic series if the first term is -7 and d = 3. n = 26, a = -7, d = 3     793 2 ( 7 ) ( 26 1 ) 3 2 26 2 ( 1 ) 2 26 26         S S a n d n S n     793 2 ( 7 ) ( 26 1 ) 3 2 26 2 ( 1 ) 2 26 26         S S a n d n S n

EXAMPLE 4 SOLUTION Find the sum of the odd numbers between 51 and 99, inclusive. a = 51, d = 2, T n = a n

T

n = a + ( n – 1) d 99 = 51 + ( n – 1) n = 25

25 25

S

S

a a

n

S

n n

25 25

S

S

a a

n

S

n n

Geometric Sequence

a1, , a 2 , a 3 ,….. , a (^) n-1, an Geometric Sequence or Geometric Progression  (^) A geometric sequence is one where each successive term is found by multiplying the preceding term by a fixed constant. 1 1   n n a a r where r is the common ratio 1 1   n n a a r

EXAMPLE

If the second term and the fifth term of a geometric sequence are ¼ and 1/32, find the first term and the common ratio.

EXAMPLE 5 SOLUTION If the second term and the fifth term of a geometric sequence are ¼ and 1/32, find the first term and the common ratio. 2 1 8 1 4 1 32 1 32 1 4 1 3 4 4 5 2         r r ar ar T ar T ar 2

2 

a a T ar

Divide (2) by (1) 2 1 8 1 4 1 32 1 32 1 4 1 3 4 4 5 2         r r ar ar T ar T ar 2

2 

a a T ar