25.1 Arithmetic sequences

When you count by 5, 6, 7, 8, 9 and onwards you would realize that you’re actually counting an ordered list of numbers. Ordered list of numbers are otherwise referred to as sequences. Consider we’re given the sequence 11, 9, 7, 5, 3, 1… If you would see, there are three dots ending the list of numbers. This indicates that this sequence is infinite and it can go on as long it could.

There are two kinds of sequences, infinite, and finite. We already saw an example of infinitesequence in the first paragraph and from there we know that they can go on and on. For the finite sequence they can just be limited with a few terms. Terms or Elements are the numbers in a sequence. The sequence has a common difference and a common ratio. This would help establish the relationship between the successive terms in the sequence. When we determine the sum of a sequence then we are able to find the series.

Apart from looking at the regular numbers like 1,2,3,4,5 and so on, we will be able to study other expressions which might be consisted of a coefficient, a variable and an exponent and other expressions such as factorials. In this chapter, we will also learn how to use mathematical induction to prove the sum of the series.

Arithmetic sequences

An arithmetic sequence (arithmetic progression) is a number sequence with a common difference between successive terms. By using the arithmetic sequence formula, we can easily find the value of a term and the common difference in the sequence.


• arithmetic sequence: a sequence with a common difference between successive terms
• The nth term, tn{t_n} ,of an arithmetic sequence:
tn=t1+(n1)d{t_n} = {t_1} + \left( {n - 1} \right)d
where, tn{t_n}: nth term
t1{t_1}: first term
dd : common difference
  • 1.
    Arithmetic sequence formula
    Consider the arithmetic sequence: 5, 9, 13, 17, … .
  • 2.
    Determine t1,d,tnt_1,d,t_n for the sequences in which two terms are given
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Arithmetic sequences

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