Chapter 8.2, Problem 97E

To determine

To write: an explanation for Gauss conclusion about sum of the consecutive natural numbers from 1 to 100. To find a formula for the sum of first n natural numbers.

Answer to Problem 97E

A formula for the sum of first n natural numbers is $\frac{n}{2}\left(n+1\right)$ .

Explanation of Solution

Calculation:

Famous mathematician Gauss saw that Sum of 1 and 100 is 101.

Sum of 2 and 99 is 101.

Sum of 3 and 98 is 101.

Continuing in the pattern Gauss concluded that the sum from 1 to 100 can be found by multiplying 101 by 50.

That is the sum will be 5050.

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d.

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

Where $a_1$ is the first term of the sequence, and d is the common difference.

Sum of an arithmetic finite sequence has the form

  $S_n=\frac{n}{2}\left(a_1+a_n\right)$

Here, n is number of terms, $a_1$ is the first term of the sequence, and $a_n$ is the last term of the sequence.

Now, first n natural number forms an arithmetic sequence with first term 1 and last term n.

Number of terms is n.

So, sum of first n natural number can be calculated as

  $S_n=\frac{n}{2}\left(1+n\right)$