Question

Let f be a function such that F(n) is the sum of the first n positive integers. Give a recursive definition of F(n).

Short answer

The recursive function definition is$F\left(1\right)=1\text{and}F\left(n\right)=F(n-1)+n\text{for}n\ge 2$

Step-by-step solution

Define the recursive sequence:

A sequence can also be defined recursively, meaning that the previous terms define successive terms in the sequence. The recursive sequence is obtained by the deriving each successive term such that it is 2 larger than the previous obtained term.

Find recursive definition of F(n)

It is given that F (n) is the sum of first n positive integers.

$F\left(n\right)=1+2+\dots \dots (n-2)+(n-1)+n$

Then,F (n - 1) is the sum of first n - 1 positive integers:

$F\left(n\right)=1+2+\dots ..(n-2)+(n-1)+n=F(n-1)+n$

Next step is to determine the first term.

The sum of the first 1 positive integer is 1 as the first positive integer is also 1.

F (1) = 1

Therefore, the recursive definition is$F\left(1\right)=1,F\left(n\right)=F(n-1)+n\text{for}n\ge 2\text{.}$.