The document discusses arithmetic sequences and series. Some key points: 1) An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This common difference (d) allows determining the nth term as an = dn + c, where c is the first term minus d. 2) The sum of the first n terms of a finite arithmetic sequence is given by S_n = (n/2)(a_1 + a_n), where a_1 is the first term and a_n is the nth term. 3) The sum of the first n terms of an infinite arithmetic sequence is called the nth partial sum. The partial sums can be represented using summation notation.

1. Arithmetic
Sequences
& Series
T- 1-855-694-8886
Email- info@iTutor.com
By iTutor.com

2. An infinite sequence is a function whose domain
is the set of positive integers.
a1, a2, a3, a4, . . . , an, . . .
The first three terms of the sequence an = 4n – 7 are
a1 = 4(1) – 7 = – 3
a2 = 4(2) – 7 = 1
a3 = 4(3) – 7 = 5.
finite sequence
terms
© iTutor. 2000-2013. All Rights Reserved

3. A sequence is arithmetic if the differences
between consecutive terms are the same.
4, 9, 14, 19, 24, . . .
9 – 4 = 5
14 – 9 = 5
19 – 14 = 5
24 – 19 = 5
arithmetic sequence
The common difference, d, is 5.
© iTutor. 2000-2013. All Rights Reserved

4. Example: Find the first five terms of the sequence
and determine if it is arithmetic.
an = 1 + (n – 1)4
This is an arithmetic sequence.
d = 4
a1 = 1 + (1 – 1)4 = 1 + 0 = 1
a2 = 1 + (2 – 1)4 = 1 + 4 = 5
a3 = 1 + (3 – 1)4 = 1 + 8 = 9
a4 = 1 + (4 – 1)4 = 1 + 12 = 13
a5 = 1 + (5 – 1)4 = 1 + 16 = 17
© iTutor. 2000-2013. All Rights Reserved

5. The nth term of an arithmetic sequence has the
form
an = dn + c
where d is the common difference and c = a1 – d.
2, 8, 14, 20, 26, . . . .
d = 8 – 2 = 6
a1 = 2 c = 2 – 6 = – 4
The nth term is 6n – 4.
© iTutor. 2000-2013. All Rights Reserved

6. a1 – d =
Example: Find the formula for the nth term of an
arithmetic sequence whose common difference is 4
and whose first term is 15. Find the first five terms
of the sequence.
an = dn + c
= 4n + 11
15,
d = 4
a1 = 15
19, 23, 27, 31.
The first five terms are
15 – 4 = 11
© iTutor. 2000-2013. All Rights Reserved

7. The sum of a finite arithmetic sequence with n
terms is given by
1( ).
2n n
nS a a
5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ?
( )501 2755 )0 5(55
2nS
n = 10
a1 = 5 a10 = 50
© iTutor. 2000-2013. All Rights Reserved

8. The sum of the first n terms of an infinite sequence
is called the nth partial sum.
1( )
2n n
nS a a
( )190 25(184) 460
2
50 6 0nS
a1 = – 6
an = dn + c = 4n – 10
Example: Find the 50th partial sum of the arithmetic
sequence – 6, – 2, 2, 6, . . .
d = 4 c = a1 – d = – 10
a50 = 4(50) – 10 = 190
© iTutor. 2000-2013. All Rights Reserved

9. The sum of the first n terms of a sequence is
represented by summation notation.
1 2 3 4
1
n
i n
i
a a a a a a
index of
summation
upper limit of summation
lower limit of summation
5
1
1
i
n (1 1) (1 2) (1 3) (1 4) (1 5)
2 3 4 5 6
20
© iTutor. 2000-2013. All Rights Reserved

10. 100
1
2
i
n
Example: Find the partial sum.
2( ) 2( ) 2( ) 2( )1 2 3 100
2 4 6 200
a1 a100
100 1 100
10( ) 2( )02 0
2 2
0nS a a
50(202) 10,100
© iTutor. 2000-2013. All Rights Reserved

11. Consider the infinite sequence a1, a2, a3, . . ., ai, . . ..
1. The sum of the first n terms of the sequence is called
a finite series or the partial sum of the sequence.
1
n
i
i
aa1 + a2 + a3 + . . . + an
2. The sum of all the terms of the infinite sequence is
called an infinite series.
1
i
i
aa1 + a2 + a3 + . . . + ai + . . .
© iTutor. 2000-2013. All Rights Reserved

12. Consider the infinite sequence a1, a2, a3, . . ., ai, . . ..
1. The sum of the first n terms of the sequence is called
a finite series or the partial sum of the sequence.
1
n
i
i
aa1 + a2 + a3 + . . . + an
2. The sum of all the terms of the infinite sequence is
called an infinite series.
1
i
i
aa1 + a2 + a3 + . . . + ai + . . .
© iTutor. 2000-2013. All Rights Reserved

13. Example: Find the fourth partial sum of
1
15 .
2
i
i
1 2 3 44
1
1 1 1 1 15 5 5 5 5
2 2 2 2 2
i
i
1 1 1 15 5 5 5
2 4 8 16
5 5 5 5
2 4 8 16
40 20 10 5 75
16 16 16 16 16
© iTutor. 2000-2013. All Rights Reserved

14. The End
Call us for more
information:
www.iTutor.com
1-855-694-8886
Visit