What is Arithmetic Sequence or Arithmetic Progression?
Arithmetic sequence (also known as arithmetic progression) is a set of integers in which the difference between consecutive terms is always the same. For example, the difference in the arithmetic sequences 1, 5, 9, 13, 17, and so on is always 4. The term for this is "common difference."
What exactly is an example of an arithmetic sequence?
An arithmetic sequence is a set of numbers in which each term has a common difference. For instance, the common difference in the arithmetic sequence 3, 9, 15, 21, 27 is 6.
Arithmetic sequence is fun to learn.
How do you find the arithmetic sequence?
To find a specific term of an arithmetic sequence, we use the formula for finding the nth term;
an = a1 +( n-1 ) * d
a1= 1 (first term in sequence)
d (difference: 2nd term - first term)
n (term number)
an (To find nth term)
What is Arithmetic Series?
The word series implies sum. We can transform a given arithmetic sequence into an arithmetic series by adding the terms of the sequence. The example below highlights the difference between the two.
Sequence versus Series
Arithmetic Sequence (list):
2,4,6,8,10,…
Arithmetic Series (sum):
2+4+6+8+10…
Notes:
▶︎ The Arithmetic Series Formula is also known as the Partial Sum Formula.
The Partial Sum Formula can be described in words as the product of the average of the first and the last terms and the total number of terms in the sum. It is fun to learn.
▶︎ The Arithmetic Sequence Formula is incorporated/embedded in the Partial Sum Formula. It is in fact the nth term or the last term
an = a1 +( n-1 ) * d
Become familiar with both the arithmetic series formula and the arithmetic sequence formula (nth term formula) because they go hand in hand when solving many problems.
Sn = n/2 [(a1 + an)]
You may recall me mentioning that the arithmetic sequence formula is embedded in the arithmetic series formula. If we substitute and expand the nth term formula within the partial sum formula, what we will get is a new and useful form of the arithmetic series formula.
Below is the alternative formula of the arithmetic series. Consider this if the last term is not given.
Sn = n/2 [2a1 + (n-1)d]
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