1. SequencesWhat is a sequence? It

1. Sequences
What is a sequence? It is a set of numbers which are written in some particular order. For
example, take the numbers
1, 3, 5, 7, 9, . . . .
Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we
start with the number 1, which is an odd number, and then each successive number is obtained
by adding 2 to give the next odd number.
Here is another sequence:
1, 4, 9, 16, 25, . . . .
This is the sequence of square numbers. And this sequence,
1, −1, 1, −1, 1, −1, . . . ,
is a sequence of numbers alternating between 1 and −1. In each case, the dots written at the
end indicate that we must consider the sequence as an infinite sequence, so that it goes on for
ever.
On the other hand, we can also have finite sequences. The numbers
1, 3, 5, 9
form a finite sequence containing just four numbers. The numbers
1, 4, 9, 16
also form a finite sequence. And so do these, the numbers
1, 2, 3, 4, 5, 6, . . . , n .
These are the numbers we use for counting, and we have included n of them. Here, the dots
indicate that we have not written all the numbers down explicitly. The n after the dots tells us
that this is a finite sequence, and that the last number is n.
Here is a sequence that you might recognise:
1, 1, 2, 3, 5, 8, . . . .
This is an infinite sequence where each term (from the third term onwards) is obtained by adding
together the two previous terms. This is called the Fibonacci sequence.
We often use an algebraic notation for sequences. We might call the first term in a sequence
u1, the second term u2, and so on. With this same notation, we would write un to represent the
n-th term in the sequence. So
u1, u2, u3, . . . , un
would represent a finite sequence containing n terms. As another example, we could use this
notation to represent the rule for the Fibonacci sequence. We would write
un = un−1 + un−2
to say that each term was the sum of the two preceding terms.

ผลลัพธ์ (อังกฤษ) 2:

1. Sequences
What is a Sequence? It is a set of numbers which are written in some particular order. For
example, take the Numbers
1, 3, 5, 7, 9,. . . .
Here, we seem to have a Rule. We have a sequence of odd numbers. To Put this another Way, we
Start with the Number 1, which is an Odd Number, and then each successive Number is obtained
by adding 2 to give the next Odd Number.
Here is another Sequence:
1, 4, 9, 16, 25. . . . .
This is the square of Sequence Numbers. And this Sequence,
1, -1, 1, -1, 1, -1,. . . ,
Is a Sequence of Numbers Alternating between 1 and -1. Case in each, the dots written at the
End indicate that we must consider as an Infinite Sequence the Sequence, so that it goes on for
Ever.
On the Other Hand, Can we also have finite sequences. The Numbers
1, 3, 5, 9
form containing just a finite Four Sequence Numbers. The Numbers
1, 4, 9, 16
also form a finite Sequence. And so do these, the Numbers
1, 2, 3, 4, 5, 6,. . . , N.
These are the Numbers Counting for we use, and we have included n of them. Here, the dots
indicate that we have not written down explicitly all the Numbers. N the dots after the US tells
that this is a finite Sequence, and that is the last Number n.
Here is a Sequence You might recognize that:
1, 1, 2, 3, 5, 8,. . . .
This is an Infinite Sequence where each term (from the third term onwards) is obtained by adding
the Together Two previous terms. Called the Fibonacci Sequence is this.
We often use an algebraic notation for sequences. The First Call we might term in a Sequence
u1, U2 the second term, and so on. Same with this notation, we would un Write to represent the
n-th term in the Sequence. So
u1, U2, U3,. . . , Un
would represent a finite Sequence containing n terms. As another example, we could use this
notation to represent the Rule for the Fibonacci Sequence. We would Write
un + 1 = un-un-2
to Say that each term was the Sum of the Two preceding terms.