Construct recurrence relations (in terms of n) for the sequences: (a) 23, 17, 11, 5, ... (b) 5, 6, 7, 8, 64, ... (c) 2, 5, 11, 23, 47, ...

What you are looking for in each list

A recurrence relation tells you how to get the next term from earlier term(s), plus an initial value like $a_1$. So we scan each sequence for a consistent “next-term rule.”

(a) Constant difference (arithmetic pattern)

Check differences:

• $17-23=-6$
• $11-17=-6$
• $5-11=-6$

Since the difference is always $-6$, the recurrence is:

• Initial term: $a_1=23$
• Recurrence: $$a_n=a_{n-1}-6, \quad n\ge 2$$

(b) “Add 1” first, then a jump to a square

From $5\to6\to7\to8$ the pattern is “add 1.” Then $8\to64$ is $8^2=64$.

A recurrence that reproduces exactly the terms shown is a piecewise recurrence:

• Initial term: $b_1=5$
• For the consecutive part: $$b_n=b_{n-1}+1, \quad 2\le n\le 4$$
• Then squaring from there: $$b_n=(b_{n-1})^2, \quad n\ge 5$$ This gives $b_2=6$, $b_3=7$, $b_4=8$, $b_5=8^2=64$ (and would continue $b_6=64^2=4096$, etc.).

(c) Multiply by 2, then add 1

Test the rule $c_n=2c_{n-1}+1$:

• $2(2)+1=5$
• $2(5)+1=11$
• $2(11)+1=23$
• $2(23)+1=47$

So the recurrence is:

• Initial term: $c_1=2$
• Recurrence: $$c_n=2c_{n-1}+1, \quad n\ge 2$$