Question

Find each sum.

1 + 3 + 5 + ... + (2n - 1)

Short answer

The sum of the given sequence is$n^2.$

Step-by-step solution

Step 1. Given information

$\text{Thegivensequenceis}1+3+5+\mathrm{...}+\text{2}\left(n-1\right)$

We have to find the sum of the sequence.

Step 2. calculating the sum

$\begin{array}{l}\text{Thesumofthearithmeticsequenceisgivenas}\\ \Rightarrow S_n=\frac{n\left(a_1+a_n\right)}{2}\\ \text{Wehave,}\\ a_1=1\\ a_n=2n-1\\ \text{Thus,}\\ \Rightarrow S_n=\frac{n\left(1+\left(2n-1\right)\right)}{2}\\ =\frac{n\left(2n\right)}{2}\\ =n^2\end{array}$