(

Example2)

Determine the rst six terms of the dened sequences, and give their associated series.

(1). f2 ng

(2). f1 + 2 n+ 3 n2

g

(3). f( 1) n

g

(4). f1 + 2 + 3 + 4 + +ng

S olutions : We denote the nth term of a sequence by a

n, and

S= a

1 +

a

2 +

a

3 +

a

4 +

a

5 +

a

6:

(1). a

n =

f2 ng

First six terms:

a 1 = 2

1 = 1 ; a

2= 2

2 = 0 ; a

3= 2

3 = 1; a

4= 2

4 = 2; a

5= 2

5 = 3; a

6= 2

6 = 4

Associated series: S= a

1 +

a

2 +

a

3 +

a

4 +

a

5 +

a

6 = 1 + 0

1 2 3 4 = 9

(2). a

n =

f1 + 2 n+ 3 n2

g

First six terms: a1 = 6

; a

2= 17

; a

3= 34

; a

4= 57

; a

5= 86

; a

6= 121

Associated series: S= a

1 +

a

2 +

a

3 +

a

4 +

a

5 +

a

6 = 6 + 17 + 34 + 57 + 86 + 121 = 321

(3). a

n =

f( 1) n

g

First six terms: a1 =

1; a

2= 1

; a

3=

1; a

4= 1

; a

5=

1; a

6= 1

Associated series: S= a

1 +

a

2 +

a

3 +

a

4 +

a

5 +

a

6 =

1 + 1 1 + 1 1 + 1 = 0

(4). a

n =

f1 + 2 + 3 + 4 + +ng

First six terms: a1 = 1

; a

2= 3

; a

3= 6

; a

4= 10

; a

5= 15

; a

6= 21

Associated series: S= a

1 +

a

2 +

a

3 +

a

4 +

a

5 +

a

6 = 1 + 3 + 6 + 10 + 15 + 21 = 56

(1). How many terms are there in an arithmetic sequence with rst term 1, common dier-

ence -3, and last term -41?

S olution :

a1 = 1

; d=3; a

n=

41 :

Find n.

an = 1 + (

n 1)( 3) = 41

1

(

n 1) =

41 1

3 = 14

)n= 15

2