# Question #b3e6b

Sep 14, 2016

$103$ toothpicks in the ${50}^{\text{th}}$ figure.
$2 n + 3$ toothpicks in the ${n}^{\text{th}}$ figure.
In each progressive figure, one toothpick is added to the top row and one is added to the bottom row. As the ${1}^{\text{st}}$ figure has $1$ toothpick in the top row and $2$ in the bottom row, that means that the ${n}^{\text{th}}$ figure will have $n$ toothpicks in the top row and $n + 1$ in the bottom row. Adding these to the $2$ toothpicks which constitute the left and right sides, we get the total for the ${n}^{\text{th}}$ figure as
$n + \left(n + 1\right) + 2 = 2 n + 3$.
To figure out how many are in the ${50}^{\text{th}}$ figure, then, we just let $n = 50$ to get $2 \left(50\right) + 3 = 103$.