An algebra problem by Lucia and Emma

The sum of the first $n$ positive odd integers is $n^2$ ,so $100^2=10000$

The sum first 100 positive odd numbers can be written as

$\displaystyle \sum_{k=1}^{100} (2k-1)$

$=\displaystyle 2\sum_{k=1}^{100} k-100$

$=2\cdot \dfrac{100 \times 101}{2} -100 = \boxed{10,000}$

$\begin{aligned} S & = 1 + 3 + 5 + \cdots + 199 & \small \color{#3D99F6} \text{Using the sum of AP: } S = \frac n2(a+l) \\ & = \frac {100}2 (1+199) & \small \color{#3D99F6} \text{where }n, a, \text{ and }l \text{ are the number of terms,} \\ & = \boxed{\text{10,000}} & \small \color{#3D99F6} \text{first, and last term of the AP respectively.} \end{aligned}$