Recurrence relationships for AP's

We have already seen that an arithmetic progression is defined a sequence of numbers where the difference between any two consecutive terms is a fixed constant $d$d. This means that each new term is generated by adding an amount $d$d to the previous term. Starting with the first term $t_1=a$t1​=a, the second term is simply given by $t_2=t_1+d$t2​=t1​+d and the third term is given by $t_3=t_2+d$t3​=t2​+d, and the fourth term is given by $t_4=t_3+d$t4​=t3​+d and so on.

This means that the $\left(n+1\right)$(n+1)th term is given by $t_{n+1}=t_n+d$tn+1​=tn​+d. For example, the $100$100th term is given by $t_{100}=t_{99}+d$t100​=t99​+d.

We thus can define an arithmetic progression with what is known as a recursive formula. We begin by indicating a first term, say $t_1=a$t1​=a, and then expressing the progression as:

$t_{n+1}=t_n+d,$tn+1​=tn​+d, with $t_1=a$t1​=a 

 So for example, the arithmetic progression given by the recursive equation $t_{n+1}=t_n+\sqrt{2}$tn+1​=tn​+√2 with $t_1=1$t1​=1 can be constructed term by term as:

                               $t_2=t_1+\sqrt{2}=1+\sqrt{2}$t2​=t1​+√2=1+√2

                               $t_3=t_2+\sqrt{2}=\left(1+\sqrt{2}\right)+\sqrt{2}=1+2\sqrt{2}$t3​=t2​+√2=(1+√2)+√2=1+2√2

                               $t_4=t_3+\sqrt{2}=\left(1+2\sqrt{2}\right)+\sqrt{2}=1+3\sqrt{2}$t4​=t3​+√2=(1+2√2)+√2=1+3√2

From the first four terms, it is clear that the sequence becomes:

$1,1+\sqrt{2},1+2\sqrt{2},1+3\sqrt{2},...,1+\left(n-1\right)\sqrt{2}+...$1,1+√2,1+2√2,1+3√2,...,1+(n−1)√2+...

An interesting sequence is given recursively as $t_{n+1}=t_n+n$tn+1​=tn​+n, with $t_1=0$t1​=0. To test whether it is an arithmetic sequence, we will consider the first four terms. Here:

                                $t_1=0$t1​=0, 

                                $t_2=t_1+2=2$t2​=t1​+2=2,

                                $t_3=t_2+3=2+3=5$t3​=t2​+3=2+3=5, 

                                $t_4=t_3+4=5+4=9$t4​=t3​+4=5+4=9, 

The differences in successive terms is not constant, but rises by $1$1 each time, so there is definitely a pattern to the sequence, but it is not arithmetic.  

Worked Examples

Question 1

Question 2

Question 3