Chapter 8.2, Problem 91E

To determine

To find: The explanation of the conclusion came and then find the formula for the sum of the first n natural number.

Answer to Problem 91E

The sum of the natural number is $S_n$ and the gauss came to conclusion by simply multiplying 50 by 101 and the summation found was 5050.

Explanation of Solution

Given:

The gauss added number 1 to 100 and found answer mentally.

Calculation:

From, the given statement the sum of the first and the last number is 101. The sum of the first and the last pattern it is concluded that there are 50 such pair of such numbers. Thus, the gauss simply multiplied 50 by 101 and the summation is 5050.

Consider the formula for the sum of the first n natural number as,

  $\begin{array}{c}{S}_n={\displaystyle \sum _{i=1}^{n}i}\\ =1+2+3+\mathrm{...}+n\\ =\frac{n}{2}\left(1+n\right)\end{array}$