The calculated series of $T(n)$ is: $T(n)= \{1,3,6,10,15,21,28,36,45,55,66\}$ for $n=0$ to $n=10$, how do you write this using sigma notation?

Question

The calculated series of $T(n)$ is: $T(n)= \{1,3,6,10,15,21,28,36,45,55,66\}$ for $n=0$ to $n=10$, how do you write this using sigma notation? $T(n)$ contains the functions:

• $T(n−1) + (n+1)$ for $n\geq1$
• $T(0) = 1$

Answer 1

So we have

\begin{align} T(0)&=1\\ T(1)&=3\\ T(2)&=6\\ T(3)&=10\\ T(4)&=15\\ T(5)&=21\\ T(6)&=28\\ T(7)&=36\\ T(8)&=45\\ T(9)&=55\\ T(10)&=66\\ \end{align} So it appears that $$ T(n)=\sum_{k=0}^n(k+1) $$

Answer 2

Not to be a smart-arse but the following is another solution:

\begin{equation} S = \sum_{t \in T(n)} t \end{equation}