Representing Numbers by areas or volumes of objects can be a great way to visually connect various problems involving numbers.
Sum of first n natural numbers is nothing but, the sum of n terms in arithmetic progression. In a general arithmetic progression with first term a and common difference d sum of n terms: S= n/2[ a + (a+( n-1)d)]
This sum has a beautiful story of origin, Carl Friedrich Gauss.
Story goes like this "In the 1780s a provincial German schoolmaster gave his class the tedious assignment of summing the first 100 integers. The teacher's aim was to keep the kids quiet for half an hour, but one young pupil almost immediately produced an answer: 1 + 2 + 3 + ... + 98 + 99 + 100 = 5,050. The smart aleck was Carl Friedrich Gauss, who would go on to join the short list of candidates for greatest mathematician ever. Gauss was not a calculating prodigy who added up all those numbers in his head. He had a deeper insight: If you "fold" the series of numbers in the middle and add them in pairs 1 + 100, 2 + 99, 3 + 98, and so on all the pairs sum to 101. There are 50 such pairs, and so the grand total is simply 50×101.
The more general formula, for a list of consecutive numbers from 1 through n, is n(n + 1)/2.
Let us find this sum visually, method for summing numbers in an arithmetic progression
We want to find the of small boxes in white colors. Same numbers of small boxes are fixed in reverse order.
These together forms a rectangle of size n*{2a +(n-1)d}. Half of the area of this rectangle is our required sum. Hence, S= n/2[ a + (a+( n-1)d)].
The reader is probably well acquainted with the following area representation of the
familiar formula for factoring the difference of two squares:
Another beautiful visual to represent the sum of squares of first n of k2 unit cubes, and compute the sum 1^2 + 2^2 +3^2+....+n^2
