{
  "translations" :{
    "en": {
      "sc1": "Let's see how to find the closed form solution for $\\displaystyle\\sum_{i=0}^{n} 2^i$.",
      "sc2": "As an example, suppose that $n = 4$.",
      "sc3.1": "When $i = 0$, this adds $1$ to the summation. Represent this by a single rectangle.",
      "sc3.2": "<br> $\\displaystyle\\sum_{i=0}^{4}2^i =$ <span style='color:red;'>1</span> $+ \\displaystyle\\sum_{i=1}^{4}2^i$",
      "sc4.1": "When $i = 1$, this adds $2$ to the summation. Represent this by two rectangles.",
      "sc4.2": "<br> $\\displaystyle\\sum_{i=0}^{4}2^i = 1 +$ <span style='color:red;'>2</span> $+ \\displaystyle\\sum_{i=2}^{4}2^i$",
      "sc5.1": "When $i = 2$, this adds $4$ to the summation. Represent this by four rectangles.",
      "sc5.2": "<br> $\\displaystyle\\sum_{i=0}^{4}2^i = 1 + 2 +$ <span style='color:red;'>4</span> $+ \\displaystyle\\sum_{i=3}^{4}2^i$",
      "sc6.1": "When $i = 3$, this adds $8$ to the summation. Represent this by eight rectangles.",
      "sc6.2": "<br> $\\displaystyle\\sum_{i=0}^{4}2^i = 1 + 2 + 4 +$ <span style='color:red;'>8</span> $+\\displaystyle\\sum_{i=4}^{4}2^i$",
      "sc7.1": "When $i = 4$, this adds $16$ to the summation. Represent this by 16 rectangles.",
      "sc7.2": "<br> $\\displaystyle\\sum_{i=0}^{4}2^i = 1 + 2 + 4 + 8 +$ <span style='color:red;'>16</span>",
      "sc8.1": "The closed form solution for this summation can be found by calculating the area of the resulting shape.",
      "sc8.2": "<br> Let's first reorder...",
      "sc12": "Now it is easy to see that the total area is $2^4 + 2^4 - 1 = 2^{4+1} - 1 = 31$.",
      "sc13": "In general, we have $\\displaystyle\\sum_{i=0}^{n} 2^i = 2^{n+1} - 1$."
    }
  }
}