Sum of n numbers Calculator

Sum of n numbers Calculator

Posted by bigboss

Sunday, September 19, 2021

Sum of Series in Javascript

Choose Sequences (K) = Σkk^2k^32k-1k(k+1)1 / k(k+1)k(k+1)(k+2)1 / k(k+1)(k+2)Number of terms (n) :

Sum of Series (Σ) :

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The triangular numbers are given by the following explicit formulas:

$${\displaystyle T_{n}=\sum _{k=1}^{n}k=1+2+3+\dotsb +n={\frac {n(n+1)}{2}}={n+1 \choose 2},}$$

where ${\displaystyle \textstyle {n+1 \choose 2}}$ is a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two".

The first equation can be illustrated using a visual proof. For every triangular number ${\displaystyle T_{n}}$, imagine a "half-square" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions ${\displaystyle n\times (n+1)}$, which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or: ${\displaystyle T_{n}={\frac {n(n+1)}{2}}}$. The example ${\displaystyle T_{4}}$ follows:

${\displaystyle 2T_{4}=4(4+1)=20}$ (green plus yellow) implies that ${\displaystyle T_{4}={\frac {4(4+1)}{2}}=10}$