the algebraic expression of a sequence first term is 1/2 and d= 1/3 is 25 belongs to this sequence
Answers
Answer:
A mathematical sequence is an ordered list of objects, often numbers. Sometimes the numbers in a sequence are defined in terms of a previous number in the list.
LEARNING OBJECTIVES
Differentiate between different types of sequences
KEY TAKEAWAYS
Key Points
The number of ordered elements (possibly infinite ) is called the length of the sequence. Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence.
An arithmetic sequence is one in which a term is obtained by adding a constant to a previous term of a sequence. So the
n
th term can be described by the formula
a
n
=
a
n
−
1
+
d
.
A geometric sequence is one in which a term of a sequence is obtained by multiplying the previous term by a constant. It can be described by the formula
a
n
=
r
⋅
a
n
−
1
.
Key Terms
sequence: An ordered list of elements, possibly infinite in length.
finite: Limited, constrained by bounds.
set: A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects that may be contained within it.
Sequences
In mathematics, a sequence is an ordered list of objects. Like a set, it contains members (also called elements or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence.
For example,
(
M
,
A
,
R
,
Y
)
is a sequence of letters that differs from
(
A
,
R
,
M
,
Y
)
, as the ordering matters, and
(
1
,
1
,
2
,
3
,
5
,
8
)
, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers
(
2
,
4
,
6
,
⋯
)
. Finite sequences are sometimes known as strings or words and infinite sequences as streams.
Examples and Notation
Finite and Infinite Sequences
A more formal definition of a finite sequence with terms in a set
S
is a function from
{
1
,
2
,
⋯
,
n
}
to
S
for some
n
>
0
. An infinite sequence in
S
is a function from
{
1
,
2
,
⋯
}
to
S
. For example, the sequence of prime numbers
(
2
,
3
,
5
,
7
,
11
,
⋯
)
is the function
1
→
2
,
2
→
3
,
3
→
5
,
4
→
7
,
5
→
11
,
⋯
A sequence of a finite length n is also called an
n
-tuple. Finite sequences include the empty sequence
(
)
that has no elements.