01204211/activity3 induction 1 - ประวัติรุ่นแก้ไข

Jittat: /* Homework 3 */

2015-09-16T04:30:54Z

Homework 3

2015-09-03T03:42:04Z

In-class activities 3

2015-09-03T02:05:56Z

Homework 3

2015-09-03T02:04:53Z

Homework 3

2015-09-03T02:03:51Z

Homework 3

2015-09-03T01:59:49Z

Homework 3

2015-09-03T01:58:23Z

2015-09-03T01:57:32Z

Homework 3

2015-09-03T01:56:33Z

In-class activities 3

2015-09-03T01:52:55Z

Homework 3

@@ แถว 34: / แถว 34: @@
 == Homework 3 ==
-: ''Due: 16 Sept 2015''
+: ''Due: 23:59 18 Sept 2015 <del>16 Sept 2015</del>''
 '''H.1''' (LPV-2.1.5) Prove the following identity:

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 03:42, 3 กันยายน 2558
แถว 25:		แถว 25:


−	'''A.5''' Prove using induction that using only 2-baht coins and 3-baht coins, one can obtain <math>n</math> baht for <math>n\geq 4</math>. (Do not use strong induction.) Also prove this statement without using induction.	+	'''A.5''' (a*) Prove using induction that using only 2-baht coins and 3-baht coins, one can obtain <math>n</math> baht for <math>n\geq 4</math>. (Do not use strong induction.)
		+
		+	(b) Also prove this statement without using induction.

@@ แถว 52: / แถว 52: @@
-'''H.5''' (R-3.3-ex-37) Show that if <math>n</math> is a positive integer then
+'''H.5*''' (R-3.3-ex-37) Show that if <math>n</math> is a positive integer then
 <center><math>\sum_{\{a_1,a_2,\ldots,a_k\}\subseteq\{1,2,\ldots,n\}} \frac{1}{a_1a_2\cdots a_k}=n</math>.</center>
 In this problem, the sum is over all nonempty subsets of <math>\{1,2,3,\ldots,n\}</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 02:04, 3 กันยายน 2558
แถว 55:		แถว 55:

	<center><math>\sum_{\{a_1,a_2,\ldots,a_k\}\subseteq\{1,2,\ldots,n\}} \frac{1}{a_1a_2\cdots a_k}=n</math>.</center>		<center><math>\sum_{\{a_1,a_2,\ldots,a_k\}\subseteq\{1,2,\ldots,n\}} \frac{1}{a_1a_2\cdots a_k}=n</math>.</center>
		+
		+	In this problem, the sum is over all nonempty subsets of <math>\{1,2,3,\ldots,n\}</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 02:03, 3 กันยายน 2558
แถว 50:		แถว 50:

	'''H.4''' (R-3.3-ex-19) Show that using only 3-baht coins and 5-baht coins, one can form a set of coins worth <math>n</math> baht for any integer <math>n > 7</math>.		'''H.4''' (R-3.3-ex-19) Show that using only 3-baht coins and 5-baht coins, one can form a set of coins worth <math>n</math> baht for any integer <math>n > 7</math>.
		+
		+
		+	'''H.5''' (R-3.3-ex-37) Show that if <math>n</math> is a positive integer then
		+
		+	<center><math>\sum_{\{a_1,a_2,\ldots,a_k\}\subseteq\{1,2,\ldots,n\}} \frac{1}{a_1a_2\cdots a_k}=n</math>.</center>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:59, 3 กันยายน 2558
แถว 49:		แถว 49:


−	'''H.4''' (R) Show that any	+	'''H.4''' (R-3.3-ex-19) Show that using only 3-baht coins and 5-baht coins, one can form a set of coins worth <math>n</math> baht for any integer <math>n > 7</math>.

@@ แถว 17: / แถว 17: @@
-'''A.3''' (MN-exercise-1b) Prove that for integer <math>n\geq 1</math>,
+'''A.3''' (MN-ex-1b) Prove that for integer <math>n\geq 1</math>,
 <center><math>\sum_{i=1}^n i\cdot 2^i = (n-1)2^{n+1} + 2.</math></center>
-'''A.4''' (R) Prove that <math>3^n < n!</math> whenever <math>n</math> is a positive integer greater than 6.
+'''A.4''' (R-3.3-ex-12) Prove that <math>3^n < n!</math> whenever <math>n</math> is a positive integer greater than 6.
@@ แถว 28: / แถว 28: @@
-'''A.6''' (MN-exercise-3a) Draw <math>n</math> lines in the plane in such a way that no two are parallel and no three intersect in a common point.  How many parts do the lines divide the plane into?  Experiment, guess the value, and prove it by induction.
+'''A.6''' (MN-ex-3a) Draw <math>n</math> lines in the plane in such a way that no two are parallel and no three intersect in a common point.  How many parts do the lines divide the plane into?  Experiment, guess the value, and prove it by induction.
 == Homework 3 ==
@@ แถว 42: / แถว 42: @@
-'''H.3''' (R-3.3-exercise6) Find the formula for
+'''H.3''' (R-3.3-ex-6) Find the formula for
 <center><math>\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)}</math>.</center>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:56, 3 กันยายน 2558
แถว 22:		แถว 22:


−	'''A.4''' (MN-exercise-3a) Draw <math>n</math> lines in the plane in such a way that no two are parallel and no three intersect in a common point. How many parts do the lines divide the plane into? Experiment, guess the value, and prove it by induction.	+	'''A.4''' (R) Prove that <math>3^n < n!</math> whenever <math>n</math> is a positive integer greater than 6.
		+
		+
		+	'''A.5''' Prove using induction that using only 2-baht coins and 3-baht coins, one can obtain <math>n</math> baht for <math>n\geq 4</math>. (Do not use strong induction.) Also prove this statement without using induction.
		+
		+
		+	'''A.6''' (MN-exercise-3a) Draw <math>n</math> lines in the plane in such a way that no two are parallel and no three intersect in a common point. How many parts do the lines divide the plane into? Experiment, guess the value, and prove it by induction.

	== Homework 3 ==		== Homework 3 ==

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:52, 3 กันยายน 2558
แถว 34:		แถว 34:

	'''H.2''' (LPV-2.5.4b) Prove that for any integer <math>n\geq 1</math>, <math>n^3-n</math> is a multiple of 6.		'''H.2''' (LPV-2.5.4b) Prove that for any integer <math>n\geq 1</math>, <math>n^3-n</math> is a multiple of 6.
		+
		+
		+	'''H.3''' (R) Find the formula for
		+
		+	<center><math>\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)}</math>.</center>
		+
		+	Try a few values of <math>n</math>, guess the formula, and prove it by induction.