What are Linear(Arithmetic) Sequences and Non-Linear Sequences
LINEAR(arithmetic) SEQUENCE is a list of numbers that increases or decreases by the same amount each time.
The amount it increases or decreases is called Term-to-Term Rule or Common Difference
Each number of a sequence is called a term.
So, the first number is a first term, second number is the second term and so on.
Linear Sequences are also named as Arithmetic Sequences.
Consider the sequence
1, 5, 9, 13, ……….
1 is the first term.
5 is the second term
9 is the third term
13 is the fourth term
and so on….
Now,
it is an increasing sequence because other than first term, every term is greater than the preceding term.
Let us find out is it a linear(arithmetic) sequence or not.
Find the difference of the term with its preceding term.
That is,
Second term – First term = 5 – 1 = 4
Third term – Second term = 9 – 5 = 4
Fourth term – Third term = 13 – 9 = 4
and so on.
Thus difference of each term with its preceding term is same, that is, 4.
So
1, 5, 9, 13, ………
is a Linear(arithmetic) Sequence whose Term-to-Term rule is ‘add 4’.
Consider another sequence
88, 80, 72, 64, ………..
It is a decreasing sequence because other than first term, every term is less than its preceding term.
80 – 88 = -8
72 – 80 = -8
64 – 72 = -8
and so on.
Thus, it is a Linear(arithmetic) Sequence, whose Term-to-Term rule is ‘subtract 8’.
NON-LINEAR SEQUENCE is a list of numbers that do not increases or decreases by the same amount.
Consider the sequence
3, 13, 27, 45, ……….
Now
13 – 3 = 10
27 – 13 = 14
45 – 27 = 18
and so on.
As the terms in the sequence are not increasing by the same amount, the sequence
3, 13, 27, 45, …….
is a non-linear sequence.
Consider the sequence
100, 80, 70, 65, ………
80 – 100 = -20
70 – 80 = -10
65 – 70 = -5
and so on.
Hence, as the terms in the sequence
100, 80, 70, 65, ……..
are not decreasing by the same amount, it is a non-linear sequence.
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