Question

 For each list of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assume the sequence starts with . Do not forget initial conditions if required.

a)     7, 11, 15, 19, 23, 27, 31, 35, 39, 43

b)    -1, 2, 1, 3, 4, 7, 11, 18, 29, 47

Answer 1

a)

$$a_2-a_1=11-7=4,a_3-a_2=15-11=4,$$

$$a_4-a_3=19-15=4,a_5-a_4=23-19=4,$$

$$a_6-a_5=27-23=4,a_7-a_6=31-27=4,$$

The sequence

$$a_8-a_7=35-31=4,a_9-a_8=39-35=4,$$

$$a_{10}-a_9=43-39=4$$

The sequence $\{a_n\}$ is arithmetic progression with $a=7, d=4.$

b)

$$a_0=-1, a_1=2,$$

$$a_3=1=-1+2=a_1+a_2,$$

$$a_4=3=2+1=a_2+a_3,$$

$$a_5=4=1+3=a_3+a_4,$$

$$a_6=7=3+4=a_4+a_5,$$

$$a_7=11=4+7=a_5+a_6,$$

$$a_8=18=7+11=a_6+a_7,$$

$$a_9=29=11+18=a_7+a_8,$$

$$a_{10}=47=18+29=a_8+a_9$$

The sequence $\{a_n\}$ satisfies the recurrence relation $a_n=a_{n-2}+a_{n-1}, n=2, 3, ...$ with $a_0=-1, a_1=2.$