Chapter 11.5, Problem 13P

a.

To determine

To explain:

Formula for $T_n$ in terms of $n$ .

Answer to Problem 13P

  $n\left(n+1\right)/2$ is formula for $T_n$ .

Explanation of Solution

Formula for $T_n$ in terms of $n$ is,

  $T_n=\frac{n\left(n+1\right)}{2}$

Where,

  $T_n=$ $n^{th}$ term,

  $n=$ Number of terms.

Therefore, $n\left(n+1\right)/2$ is formula for $T_n$ .

b.

To determine

To calculate:

Formula of $T_n$ using sigma notation.

Answer to Problem 13P

Formula of $T_n$ using sigma notation is ${\displaystyle {\sum }_{k=1}^{n}k}$ .

Explanation of Solution

Formula for $T_n$ in terms of $n$ is,

  $\begin{array}{l}{T}_n={\displaystyle {\sum }_{k=1}^{n}k=1+2+3+\mathrm{...}+n}\\ T_n=\frac{n\left(n+1\right)}{2}\end{array}$

Where,

  $T_n=$

  $n^{th}$ term,

  $n=$ Number of terms.

  $\frac{n\left(n+1\right)}{2}=$ binomial coefficient

Therefore, ${\displaystyle {\sum }_{k=1}^{n}k}$ is Formula of $T_n$ using sigma notation.