Add example Proof Blocks problem

Author: SethPoulsen

runestone/parsons/test/_sources/index.rst

@@ -113,3 +113,47 @@ Section 2: Labeling
    =====
    print(sum)
 
+
+============
+Proof Blocks
+============
+
+.. raw:: html
+
+    <embed>
+      <!-- TODO load dependencies properly through NPM -->
+      <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+      <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+    </embed>
+
+.. parsonsprob:: test_parsons_4
+   :language: math
+   :grader: dag
+
+   .. raw:: html
+
+    <embed>   Drag and drop <strong style="color:red;">all</strong> of the blocks below to create a proof by induction of the following statement.
+      Claim: for all natural numbers \(n\),
+      $$
+      \sum_{j=0}^n 2(-7)^j = \frac{1-(-7)^{n+1}}{4}
+      $$
+    </embed>
+
+   -----
+   Proof by induction on \(n\). #tag:0;depends:;
+   =====
+   Inductive Predicate:  \(P(n): \sum_{j=0}^n 2(-7)^j = \frac{1-(-7)^{n+1}}{4}\) #tag:1;depends:0;
+   =====
+   Base case: At \(n=0\), \(\sum_{j=0}^n 2(-7)^j = 2\) and \(\frac{1-(-7)^{n+1}}{4} = \frac{1-(-7)}{4} = 2\), so the base case, \(P(0)\), holds #tag:2;depends:0,1;
+   =====
+   Inductive Hypothesis: Suppose that \(P(n):\) \(\sum_{j=0}^n 2(-7)^j = \frac{1-(-7)^{n+1}}{4}\) holds for \(n=0,1,...,k\). #tag:3;depends:1;
+   =====
+   Inductive Step: We need to show that \(P(k+1):\) \(\sum_{j=0}^{k+1} 2(-7)^j = \frac{1-(-7)^{k+2}}{4}\)  holds #tag:4;depends:3;
+   =====
+   The left hand side is \(\sum_{j=0}^{k+1} 2(-7)^j = \sum_{j=0}^k 2(-7)^j + 2(-7)^{k+1}\) #tag:5;depends:4;
+   =====
+   By the inductive hypothesis we have \(\sum_{j=0}^k 2(-7)^j = \frac{1-(-7)^{k+1}}{4}\). So then substituting we get \(= \frac{1-(-7)^{k+1}}{4} + 2(-7)^{k+1}\) \(= \frac{1-(-7)^{k+1} + 8(-7)^{k+1}}{4}\) \(= \frac{1+7(-7)^{k+1}}{4}\) which simplifies to \(= \frac{1-(-7)^{k+2}}{4}\) #tag:6;depends:5;
+   =====
+   So \(\sum_{j=0}^{k+1} 2(-7)^j = \frac{1-(-7)^{k+2}}{4}\), which was what we needed to show. #tag:7;depends:6;
+
+