{
  "translations" :{
    "en": {
      "sc1": "Let's see how to find the closed form solution for $\\displaystyle\\sum_{i=1}^{n}i$.",
      "sc2": "As an example, suppose that $n = 5$.",
      "sc3.1": "When $i=1$, this adds $1$ to the summation. Represent this by a single rectangle.",
      "sc3.2": "<br> $\\displaystyle\\sum_{i=1}^{n=5}i =$ <span style='color:red;'>1</span> $+ \\displaystyle\\sum_{i=2}^{n=5}i.$",
      "sc4.1": "When $i = 2$, this adds $2$ to the overall summation. Represent this by two rectangles.",
      "sc4.2": "<br> $\\displaystyle\\sum_{i=1}^{n=5}i = 1 +$ <span style='color:red;'>2</span> $+ \\displaystyle\\sum_{i=3}^{n=5}i$",
      "sc5.1": "When $i = 3$, this adds $3$ to the summation. Represent this by three rectangles.",
      "sc5.2": "<br> $\\displaystyle\\sum_{i=1}^{n=5}i = 1 + 2 +$ <span style='color:red;'>3</span> $+ \\displaystyle\\sum_{i=4}^{n=5}i$",
      "sc6.1": "When $i = 4$, this adds $4$ to the summation. Represent this by four rectangles.",
      "sc6.2": "<br> $\\displaystyle\\sum_{i=1}^{n=5}i = 1 + 2 + 3 +$ <span style='color:red;'>4</span> $+ \\displaystyle\\sum_{i=5}^{n=5}i$",
      "sc7.1": "When $i = 5$, this adds $5$ to the summation. Represent this by five rectangles.",
      "sc7.2": "<br> $\\displaystyle\\sum_{i=1}^{n=5}i = 1 + 2 + 3 + 4 +$ <span style='color:red;'>5</span>",
      "sc8": "The closed form solution for this summation can be found by calculating the area of the resulting shape.",
      "sc9": "The area will be the area of the big triangle",
      "sc10" : "$\\ +$ the sum of the areas of the $n$ small rectangles.",
      "sc11": "So, the area is $\\displaystyle\\frac{n^2}{2} + \\frac{n}{2} = \\frac{n(n + 1)}{2}$.",
      "sc12": "Finally, we have $\\displaystyle\\sum_{i=1}^{n} i = \\frac{n(n + 1)}{2}$."
    }
  }
}