This is a $3$ dimensional analogous to Gauss's solution.We have in fact conveniently rearrange the numbers this time with the help of a $3$ dimensional space visualization to get a sum where all the numbers are equal to in this case $T_n$. It tells us that we are summing something. Unpacking the meaning of summation notation This is the sigma symbol. The Greek capital letter, sigma, is used to express long sums of values in a compact form. Summation notation (or sigma notation) allows us to write a long sum in a single expression. To make it easier to write down these lengthy sums, we look at some new notation here, called sigma notation (also known as summation notation). Learn how to evaluate sums written this way. In the sense that they still bear the same discrete numbers. Summation notation We can describe sums with multiple terms using the sigma operator. But in this case deformations that keep the numbers of balls by layers the same. It's easier to visualize this if you think of these figures as continuous figures as a tetrahedra a pyramid and the wage and we are able to transform them using a principle similar to the cavalieri principle for volume of figures. A sum in sigma notation looks something like this: The (sigma) indicates that a sum. So we could say from k equals 0 all the way to k equals n of a times r to the k-th power. Sigma notation is a way of writing a sum of many terms, in a concise form. We have seen the prior relation relating two tetrahedral to a triangular and a pyramidal so is natural to think on applying that relation here so from the $3$ tetrahedral one is lead to one tetrahedral plus one pyramidal plus one triangular.Ĭan you see now $3$ dimensionally how to get from there to the prism $(n 2) T_n$ ? So we can write this as the sum, so capital sigma right over here. It is immediate to see this geometrically as $3$ tetrahedral and the $(n 2) T_n$ as a prism or wedge with triangular base. There is a very beautiful demonstration of this fact in the book of numbers by Conway and Guy but I will not reproduce that here instead I will show that this relation is true thinking $3$ dimensionally. So we are conjecturing that this relation is true.$$3 Tetr (n) = (n 2) T_n$$ The more elements one place the more tuned the answer will be. $3$ times tetrahedral $n$ equals $(n 2)T_n$also there is a web site at where one is able to place a few elements of a sequence and one can get what known sequences contain those elements. After computing a few tetrahedral numbers$1,4,10,20.$ one is lead to suspect that the following relation is true. We set out to find a formula for the pyramidal $n$ and now we have in hand a relation that involve the tetrahedral numbers so let us find a formula for them now. That is a very nice relation that links triangular numbers, tetrahedral numbers and pyramidal numbers. Then we can translate our prior relation into $$2 Tetr (n)=T_n Pyr(n)$$ The sum $1 2 3 \cdots n$ could be seen as $1^1 2^1 3^1 \cdots n^1$ and that immediately suggests We talked about generalizations as a way to discover new results in mathematics.
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