For a sequence that is given by the explicit rule
an=a1⋅np,
where p is an integer, a1 and p can be found by calculating the differences between consecutive terms of the sequence.
Consider the sequence
First it is necessary to calculate the differences between the terms. Here there differences are as follows.
an | an−1 | an−an−1 | Difference |
---|---|---|---|
208 | 121.5 | 288−121.5 | 166.5 |
562.5 | 288 | 562.5−288 | 274.5 |
972 | 562.5 | 972−562.5 | 409.5 |
1543.3 | 972 | 1543.5−972 | 571.5 |
A new sequence can then be created using these differences.
The integer p is the same as the number of differences needed. Here, since the 3rd the differences were calculated gave identical results p equals 3.
a1 can be calculated by using the relationship d=a1⋅p⋅(p−1)⋅…⋅1,
where d is the number on row number p.
Here, we have d=27 and p=3.
27=a1⋅3⋅2⋅⋅1⇔a1=4.5