fix ch4 typos

Author: oscarlevin

source/ch_sequences.ptx

@@ -4,7 +4,7 @@
   <title>Sequences</title>
   <introduction>
     <p>
-      We have encountered <em>finite</em> sequences already as a discrete structure and something we can count.  In this chapter, we will consider possibly <em>infinite</em> sequences of numbers.  When the sequence itself is infinite, it no longer makes sense to ask how many possible sequences there are, but there is still an interesting connection to counting: each term in the sequence can represent an answer to a counting question!  
+      We have encountered <em>finite</em> sequences already as a discrete structure and something we can count.  In this chapter, we will consider possibly <em>infinite</em> sequences of numbers.  When the sequence itself is infinite, it no longer makes sense to ask how many possible sequences there are, but there is still an interesting connection to counting: Each term in the sequence can represent an answer to a counting question!  
     </p>
 
     <!-- <investigation>

source/exercises/seq-basics.ptx

@@ -314,7 +314,7 @@
     <idx><h>ternary string</h></idx>
     <statement>
       <p>
-        A <term>ternary</term> string is a sequence of 0's, 1's and 2's.  Just like a bit string, but with three symbols.  
+        A <term>ternary</term> string is a sequence of 0's, 1's, and 2's.  Just like a bit string, but with three symbols.  
       </p>
 
       <p>
@@ -393,7 +393,7 @@
         The queen bee can move one space at a time either directly to the right or angled up-right or down-right
         (but can never move leftwards).
         How many different paths can the queen take from the top left hexagon to the bottom right hexagon?
-        Explain your answer, and this relates to the previous question.
+        Explain your answer, and how this relates to the previous question.
         (As an example,
         there are three paths to get to the second hexagon on the bottom row.)
       </p>
@@ -439,7 +439,7 @@
   <idx><h>domino</h></idx>
     <statement>
       <p>
-        Let <m>t_n</m> denote the number of ways to tile a <m>2\times n</m> chessboard using <m>1\times 2</m> dominoes.  Write out the first few terms of the sequence <m>(t_n)_{n \ge 1}</m> and then give a recursive definition.  Explain why your recursive formula is correct. 
+        Let <m>t_n</m> denote the number of ways to tile a <m>2\times n</m> chessboard using <m>1\times 2</m> dominoes.  Write out the first few terms of the sequence <m>(t_n)_{n \ge 1}</m>, and then give a recursive definition.  Explain why your recursive formula is correct. 
       </p>
     </statement>
     <hint>

source/exercises/seq-conc.ptx

@@ -41,7 +41,7 @@
           <li>
             <p>
               Find a closed formula once again,
-              this time by recognizing the sequence as a modification to some well-known sequence(s).
+              this time by recognizing the sequence as a modification of some well-known sequence(s).
               Explain.
             </p>
           </li>
@@ -63,13 +63,13 @@
         <ol>
           <li>
             <p>
-              The sequence of partial sums.
+              The sequence of partial sums?
             </p>
           </li>
 
           <li>
             <p>
-              The sequence of second differences.
+              The sequence of second differences?
             </p>
           </li>
         </ol>
@@ -99,7 +99,7 @@
     <statement>
       <p>
         Consider the sequence given recursively by <m>a_1 = 4</m>,
-        <m>a_2 = 6</m> and <m>a_n = a_{n-1} + a_{n-2}</m>.
+        <m>a_2 = 6</m>, and <m>a_n = a_{n-1} + a_{n-2}</m>.
 
         <ol>
           <li>
@@ -144,7 +144,7 @@
         <statement>
           <p>
             The sequence <m>(a_n)_{n \ge 1}</m> starts
-            <m>-1, 0, 2, 5, 9, 14\ldots</m> and has closed formula <m>a_n = \dfrac{(n+1)(n-2)}{2}</m>.
+            <m>-1, 0, 2, 5, 9, 14\ldots</m> and has closed formula <me>a_n = \dfrac{(n+1)(n-2)}{2}</me>.
             Use this fact to find a closed formula for the sequence
             <m>(b_n)_{n \ge 1}</m> which starts <m>4, 10, 18, 28, 40, \ldots</m>.
           </p>
@@ -165,9 +165,9 @@
     <idx><h>Christmas</h></idx>
     <statement>
       <p>
-        The in song <em>The Twelve Days of Christmas</em>,
-        my true love gave to me first 1 gift,
-        then 2 gifts and 1 gift, then 3 gifts, 2 gifts and 1 gift,
+        In the song <em>The Twelve Days of Christmas</em>,
+        my true love gave to me first 1 gift;
+        then 2 gifts and 1 gift; then 3 gifts, 2 gifts, and 1 gift;
         and so on.
         How many gifts did my true love give me all together during the twelve days?
       </p>
@@ -203,7 +203,7 @@
 
           <li>
             <p>
-              Give a recursive definition of the sequence and explain why it is correct.
+              Give a recursive definition of the sequence, and explain why it is correct.
             </p>
           </li>
 
@@ -258,11 +258,11 @@
     <idx><h>sequence of partial sums</h></idx>
     <statement>
       <p>
-        Consider the sequence of partial sums of <em>squares</em> of Fibonacci numbers: <m>F_1^2</m>, <m>F_1^2 + F_2^2</m>, <m>F_1^2 + F_2^2 + F_3^2, \ldots</m>.  The sequences starts <m>1, 2, 6, 15, 40,\ldots</m> 
+        Consider the sequence of partial sums of <em>squares</em> of Fibonacci numbers: <m>F_1^2</m>, <m>F_1^2 + F_2^2</m>, <m>F_1^2 + F_2^2 + F_3^2, \ldots</m>.  The sequence starts <m>1, 2, 6, 15, 40,\ldots</m> 
         <ol>
           <li>
             <p>
-              Guess a formula for the <m>n</m>th partial sum, in terms of Fibonacci numbers.  Hint: write each term as a product.
+              Guess a formula for the <m>n</m>th partial sum, in terms of Fibonacci numbers.  Hint: Write each term as a product.
             </p>
           </li>
           <li>

source/exercises/seq-exponential.ptx

@@ -112,7 +112,7 @@
 
           <li>
             <p>
-              Check your solution for the closed formula by solving the recurrence relation using the Characteristic Root technique.
+              Check your solution for the closed formula by solving the recurrence relation using the characteristic root technique.
             </p>
           </li>
         </ol>
@@ -198,7 +198,7 @@
 
           <li>
             <p>
-              Solve the recurrence relation using the Characteristic Root technique.
+              Solve the recurrence relation using the characteristic root technique.
             </p>
           </li>
         </ol>

source/exercises/seq-growth.ptx

@@ -33,7 +33,7 @@
       <p>
         Is there a pair of integers <m>(a,b)</m> such that
         <m>a,
-        x_1, y_1, b</m> is part of an arithmetic sequences and
+        x_1, y_1, b</m> is part of an arithmetic sequence and
         <m>a,
         x_2, y_2, b</m> is part of a geometric sequence with <m>x_1, x_2, y_1, y_2</m> all integers?
       </p>
@@ -93,13 +93,13 @@
               Create a sequence of rectangles using this rule starting with a <m>1\times 2</m> rectangle.
               Then write out the sequence of <em>perimeters</em> for the rectangles
               (the first term of the sequence would be 6, since the perimeter of a
-              <m>1\times 2</m> rectangle is 6 - the next term would be 10).
+              <m>1\times 2</m> rectangle is 6; the next term would be 10).
             </p>
           </li>
 
           <li>
             <p>
-              Repeat the above part this time starting with a <m>1 \times 3</m> rectangle.
+              Repeat the above part, this time starting with a <m>1 \times 3</m> rectangle.
             </p>
           </li>
 

source/exercises/seq-induction.ptx

@@ -11,7 +11,7 @@
         you exchange your cow for some magic dark chocolate espresso beans.
         These beans have the property that every night at midnight,
         each bean splits into two, effectively doubling your collection.
-        You decide to take advantage of this and each morning (around 8 am) you eat 5 beans.
+        You decide to take advantage of this, and each morning (around 8 am) you eat 5 beans.
 
         <ol>
           <li>
@@ -294,7 +294,7 @@
       <p>
         Zombie Euler and Zombie Cauchy,
         two famous zombie mathematicians,
-        have just signed up for <delete>Twitter</delete> X accounts.
+        have just signed up for Myspace accounts.
         After one day, Zombie Cauchy has more followers than Zombie Euler.
         Each day after that,
         the number of new followers of Zombie Cauchy is exactly the same as the number of new followers of Zombie Euler
@@ -310,7 +310,7 @@
     <idx><h>football (American)</h></idx>
     <statement>
       <p>
-        Find the largest number of points that a football team cannot get exactly using just 3-point field goals and 7-point touchdowns
+        Find the largest number of points that it is impossible for a football team to get exactly, using just 3-point field goals and 7-point touchdowns
         (ignore the possibilities of safeties,
         missed extra points, and two-point conversions).
         Prove your answer is correct by mathematical induction.
@@ -362,7 +362,7 @@
         <p>
           Let <m>P(n)</m> be the statement that <m>n + 3 = n + 7</m>.
           We will prove that <m>P(n)</m> is true for all <m>n \in \N</m>.
-          Assume, for induction that <m>P(k)</m> is true.
+          Assume, for induction, that <m>P(k)</m> is true.
           That is, <m>k+3 = k+7</m>.
           We must show that <m>P(k+1)</m> is true.
           Now since <m>k + 3 = k + 7</m>, add 1 to both sides.
@@ -538,7 +538,7 @@
         Prove that there is a sequence of positive real numbers
         <m>a_0, a_1, a_2, \ldots</m> such that the partial sum
         <m>a_0 + a_1 + a_2 + \cdots + a_n</m> is strictly less than <m>2</m> for all <m>n \in \N</m>.
-        Hint: think about how you could define what <m>a_{k+1}</m> is to make the induction argument work.
+        Hint: Think about how you could define what <m>a_{k+1}</m> is to make the induction argument work.
       </p>
     </statement>
     <hint>

source/exercises/seq-polynomial.ptx

@@ -181,7 +181,7 @@
   <exercise>
     <statement>
       <p>
-        If you were to shade in a <m>n\times n</m> square on graph paper,
+        If you were to shade in an <m>n\times n</m> square on graph paper,
         you could do it the boring way
         (with sides parallel to the edge of the paper)
         or the interesting way, as illustrated below:
@@ -236,7 +236,7 @@
       <p>
         The interesting thing here is that a
         <m>3\times 3</m> square now has area 13.
-        Our goal is the find a formula for the area of a <m>n \times n</m> (diagonal) square.
+        Our goal is to find a formula for the area of an <m>n \times n</m> (diagonal) square.
 
         <ol>
           <li>
@@ -377,9 +377,9 @@
       </sidebyside>
 
       <p>
-        Note, these are solid tetrahedrons,
+        Note: These are solid tetrahedrons,
         so there will be some cannonballs obscured from view (the picture on the right has one cannonball in the back not shown in the picture,
-        for example)
+        for example).
       </p>
 
       <p>
@@ -392,7 +392,7 @@
             <p>
               Let <m>P(n)</m> denote the number of cannonballs needed to create a pyramid <m>n</m> layers high.
               So <m>P(1) = 1</m>, <m>P(2) = 4</m>, and so on.
-              Calculate <m>P(3)</m>, <m>P(4)</m> and <m>P(5)</m>.
+              Calculate <m>P(3)</m>, <m>P(4)</m>, and <m>P(5)</m>.
             </p>
           </li>
 
@@ -406,7 +406,7 @@
           <li>
             <p>
               Answer the pirate's question:
-              how many cannonballs do they need to make a pyramid 15 layers high?
+              How many cannonballs do they need to make a pyramid 15 layers high?
             </p>
           </li>
 

source/exercises/seq-strong-induction.ptx

@@ -11,7 +11,7 @@
         Suppose a football team only scores 3-point field goals and 7-point touchdowns
         (ignore the possibilities of safeties,
         missed extra points, and two-point conversions).
-        Prove, using <em>strong</em> induction, that the team can get any number of points 12 or greater. 
+        Prove, using <em>strong</em> induction, that the team can get any number of points, 12 points or greater. 
       </p>
     </statement>
     <hint>
@@ -43,7 +43,7 @@
     <idx><h>binary representation</h></idx>
     <statement>
       <p>
-        Prove that every positive integer is either a power of 2, or can be written as the sum of distinct powers of 2.
+        Prove that every positive integer is either a power of 2 or can be written as the sum of distinct powers of 2.
       </p>
     </statement>
     <solution>
@@ -163,16 +163,16 @@
         <task>
           <statement>
             <p>
-              First fix <m>m = 0</m> and give a proof by mathematical induction that <m>P(0,n)</m> holds for all <m>n \ge 0</m>.
-              Note this proof will be very easy.
+              First fix <m>m = 0</m>, and give a proof by mathematical induction that <m>P(0,n)</m> holds for all <m>n \ge 0</m>.
+              Note that this proof will be very easy.
             </p>
           </statement>
         </task>
 
           <task>
             <statement>
             <p>
-              Now fix an arbitrary <m>n</m> and give a proof by <em>strong</em>
+              Now fix an arbitrary <m>n</m>, and give a proof by <em>strong</em>
               mathematical induction that <m>P(m,n)</m> holds for all <m>m \ge 0</m>.
             </p>
           </statement>