When we are given a sequence, generally we see its first few terms.
a_1, a_2, a_3, a_4, ...
To find the next terms we can reason inductively. To do this can follow several strategies, like finding a relationship between consecutive terms. Let's examine two examples.
Note that consecutive terms in Sequence (I) have a common difference of 1.5. Conversely, consecutive terms in Sequence (II) have a common ratio of 0.5. To find the rule for the nth term, we can rewrite each term in terms of the common difference or the common ratio. Let's do it for Sequence (I).
Term
Value
In Terms of the Common Difference
a_1
1
1.5( 1)-0.5
a_2
2.5
1.5( 2)-0.5
a_3
4
1.5( 3)-0.5
...
...
...
a_n
-
1.5 n-0.5
This final equation shows the rule for the nth term of Sequence (I). In a similar way, we can find the rule for Sequence II.
Terms
Rule
Sequence I
1,2.5,4,5.5,7, ...
a_n=1.5n-0.5
Sequence II
8,4,2,1,1/2, ...
b_n=8(1/2)^(n-1)
However, there are sequences in which the relationship between the terms cannot be explained in the same way. When this is the case we look for other recognizable patterns. Let's investigate another sequence.
1/4, 4/4, 9/4, 16/4, 25/4, ...
Note that the denominator of all the terms in the above sequence is 4. Moreover, the numerator is the square of the term's position in the sequence. Considering this, we can write a rule for the nth term.
a_n = n^2/4
As a result, to write the rule for the nth term of a sequence we use inductive reasoning to recognize any visible pattern. Then, we use this pattern to write the rule.
