Q. 9 - Calculus

Question

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

Step-by-Step Solution

Answer

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

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 a 1 = 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 $k\ge 2$.