Chapter F, Problem 16E

To determine

To write: The sum $1+3+5+7+\cdots +\left(2n-1\right)$ in sigma notation.

Answer to Problem 16E

The sum $1+3+5+7+\cdots +\left(2n-1\right)$ expressed in sigma notation as ${\displaystyle \sum _{i=1}^n\left(2i-1\right)}$.

Explanation of Solution

Definition used:

If $a_m,a_{m+1},\mathrm{...},a_n$ are real numbers and m and n are integers such that $m\le n$, then ${\displaystyle \sum _{i=m}^n{a}_i=a_m+}{a}_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_n$.

Calculation:

Here, the sum $1+3+5+7+\cdots +\left(2n-1\right)$ is the sum of $n$ numbers and number is of the form $2i-1$, where $i$ varies from 1 to $n$.

Thus, by the above definition the sum $1+3+5+7+\cdots +\left(2n-1\right)$ expressed as follows.

$1+3+5+7+\cdots +\left(2n-1\right)={\displaystyle \sum _{i=1}^n\left(2i-1\right)}$

Thus, the sum $1+3+5+7+\cdots +\left(2n-1\right)$ expressed in sigma notation as ${\displaystyle \sum _{i=1}^n\left(2i-1\right)}$.