(
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

(1). how many terms are there in an arithmetic sequence with rst term 1, common dierence
-3, and last term -41?
S olution :
a1 = 1
; d=3; a
n=
41 :
Find n.
an = 1 + (
n 1)( 3) = 41
( n 1) =
41 1
3 = 14
😉 n= 15 :
2

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