Question

Give a recursive definition for the sequence $1,2,3,4,....$of positive integers. (Hint: Let $a_1=1$.)

Short answer

The recursive definition for the sequence$1,2,3,4,....$ is$a_k=a_{k-1}+1$where$k\ge 2$.

Step-by-step solution

Step 1. Given information

The given sequence is,

$1,2,3,4,...$of positive integers.

Step 2. We know that sequence of the consecutive positive integers with first term a1=1.

Now, the second term,

$$\begin{gathered} a_2=2 \\ =1+1 \\ =a_1+1 \end{gathered}$$

Hence, $a_2=a_1+1$.

Similarly, $a_3=a_2+1$.

Continuing, $a_k=a_{k-1}+1$.

Hence, the recursive definition for the sequence is,$a_k=a_{k-1}+1$where$k\ge 2$.