Question

Find at least three different sequences beginning with the terms 3, 5, 7 whose terms are generated by a simple for- mula or rule.

Answer 1

1. Perhaps the simplest rule is one that makes this the start of an arithmetic sequence:

$a_{n+1}=a_{n}+2$

Then the sequence will continue ...

... 3, 5, 7, 9, 11, 13, 15, ...

2. We can multiply the two terms by some values, add the results, then add some constant. There are an infinite number of ways to choose such values. Here's one set of numbers that give the third term from the first two:

$a_{n+2}=4a_n-a_{n+1}$

Then the sequence will continue ...

... 3, 5, 7, 13, 15, 37, 23, ...

3. We can multiply adjacent values and add a constant.

$a_{n+2}=a_{n+1}\cdot a_n-8$

Then the sequence will continue ...

... 3, 5, 7, 28, 188, 5256, 988120, ...