document.write( "Question 257972: bob made the following conjecture:if you add three consecutive integers,then their sum is equivalent to three times the secound integer.\r
\n" ); document.write( "\n" ); document.write( "A.give 4 examples of 3 consecutive integers.
\n" ); document.write( "B. test bob's conjecture for the integers in part a.
\n" ); document.write( "C.give either a counterexample or a deductive proof of bob's conjecture. \n" ); document.write( "
Algebra.Com's Answer #189943 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
bob made the following conjecture:if you add three consecutive integers,then their sum is equivalent to three times the secound integer.\r
\n" ); document.write( "\n" ); document.write( "A.give 4 examples of 3 consecutive integers.
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document.write( "Learn that:\r\n" );
document.write( "\"3 consecutive integers\" means \"3 whole numbers in a row\"\r\n" );
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document.write( "One example of three whole numbers is a row is 5,6 and 7\r\n" );
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document.write( "Another example of 3 whole numbers in a row is 20, 21, and 22.\r\n" );
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document.write( "Another example of 3 whole numbers in a row is 1, 2, and 3.\r\n" );
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document.write( "Another example of 3 whole numbers in a row is 14,15, and 16.\r\n" );
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\n" ); document.write( "B. test bob's conjecture for the integers in part a.
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document.write( "In the first example above, 5, 6 and 7, \r\n" );
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document.write( "the first one is 5, the second one is 6, and the third one is 7.\r\n" );
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document.write( "Now let's see what Bob said about those.\r\n" );
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document.write( "His first 6 words are:\r\n" );
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document.write( "\"if you add three consecutive integers,...\"\r\n" );
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document.write( "So let's stop and do that. We will add 5+6+7.  We get 18.\r\n" );
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document.write( "Let's see some more of what Bob said:\r\n" );
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document.write( "\"then their sum...\" \r\n" );
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document.write( "\"Their sum\" means what we got when we added them, which was 18.\r\n" );
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document.write( "\"is equivalent to...\"\r\n" );
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document.write( "that means \"is equal to\"\r\n" );
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document.write( "\"three times the second integer.\"\r\n" );
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document.write( "The second integer is 6.  So lets multiply 3 times 6. We get 18.\r\n" );
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document.write( "Now we ask?  Did we get 18 both times?  The answer is yes.\r\n" );
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document.write( "In the second example above, 20, 21 and 22, \r\n" );
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document.write( "the first one is 20, the second one is 21, and the third one is 22.\r\n" );
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document.write( "Now let's see what Bob said about those.\r\n" );
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document.write( "His first 6 words are:\r\n" );
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document.write( "\"if you add three consecutive integers,...\"\r\n" );
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document.write( "So let's stop and do that. We will add 20+21+22.  We get 63.\r\n" );
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document.write( "Let's see some more of what Bob said:\r\n" );
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document.write( "\"then their sum...\" \r\n" );
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document.write( "\"Their sum\" means what we got when we added them, which was 63.\r\n" );
document.write( "\r\n" );
document.write( "\"is equivalent to...\"\r\n" );
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document.write( "that means \"is equal to\"\r\n" );
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document.write( "\"three times the second integer.\"\r\n" );
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document.write( "The second integer is 21.  So lets multiply 3 times 21. We get 63.\r\n" );
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document.write( "Now we ask:  Did we get 63 both times?  The answer is yes.\r\n" );
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document.write( "In the third example above, 1, 2 and 3, \r\n" );
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document.write( "the first one is 1, the second one is 2, and the third one is 3.\r\n" );
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document.write( "Now let's see what Bob said about those.\r\n" );
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document.write( "His first 6 words are:\r\n" );
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document.write( "\"if you add three consecutive integers,...\"\r\n" );
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document.write( "So let's stop and do that. We will add 1+2+3.  We get 6.\r\n" );
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document.write( "Let's see some more of what Bob said:\r\n" );
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document.write( "\"then their sum...\" \r\n" );
document.write( "\r\n" );
document.write( "\"Their sum\" means what we got when we added them, which was 6.\r\n" );
document.write( "\r\n" );
document.write( "\"is equivalent to...\"\r\n" );
document.write( "\r\n" );
document.write( "that means \"is equal to\"\r\n" );
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document.write( "\"three times the second integer.\"\r\n" );
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document.write( "The second integer is 2.  So lets multiply 3 times 2. We get 6.\r\n" );
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document.write( "Now we ask:  Did we get 6 both times?  The answer is yes.\r\n" );
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document.write( "In the fourth example above, 14, 15 and 16, \r\n" );
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document.write( "the first one is 14, the second one is 15, and the third one is 16.\r\n" );
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document.write( "Now let's see what Bob said about those.\r\n" );
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document.write( "His first 6 words are:\r\n" );
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document.write( "\"if you add three consecutive integers,...\"\r\n" );
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document.write( "So let's stop and do that. We will add 14+15+16.  We get 45.\r\n" );
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document.write( "Let's see some more of what Bob said:\r\n" );
document.write( "\r\n" );
document.write( "\"then their sum...\" \r\n" );
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document.write( "\"Their sum\" means what we got when we added them, which was 45.\r\n" );
document.write( "\r\n" );
document.write( "\"is equivalent to...\"\r\n" );
document.write( "\r\n" );
document.write( "that means \"is equal to\"\r\n" );
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document.write( "\"three times the second integer.\"\r\n" );
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document.write( "The second integer is 15.  So let's multiply 3 times 15. We get 45.\r\n" );
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document.write( "Now we ask:  Did we get 45 both times?  The answer is yes.\r\n" );
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\n" ); document.write( "C.give either a counterexample or a deductive proof of bob's conjecture.
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document.write( "We think his comnjecture is true because our answer was \"yes\" every time\r\n" );
document.write( "in those four examples.\r\n" );
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document.write( "Let the three consecutive integers, whole numbers in a row\r\n" );
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document.write( "be N, N+!, and N+2\r\n" );
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document.write( "The first one is N, the second one is N+1, and the third one is N+2.\r\n" );
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document.write( "Now let's see what Bob said about those.\r\n" );
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document.write( "His first 6 words are:\r\n" );
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document.write( "\"if you add three consecutive integers,...\"\r\n" );
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document.write( "So let's stop and do that. We will add \r\n" );
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document.write( "N + (N+1) + (N+2)\r\n" );
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document.write( "Removing the parentheses:\r\n" );
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document.write( "N + N + 1 + N + 2\r\n" );
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document.write( "We will combine like terms:\r\n" );
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document.write( "3N + 3 \r\n" );
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document.write( "We get 3N + 3.\r\n" );
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document.write( "Let's see some more of what Bob said:\r\n" );
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document.write( "\"then their sum...\" \r\n" );
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document.write( "\"Their sum\" means what we got when we added them, which was 3N + 3.\r\n" );
document.write( "\r\n" );
document.write( "\"is equivalent to...\"\r\n" );
document.write( "\r\n" );
document.write( "that means \"is equal to\"\r\n" );
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document.write( "\"three times the second integer.\"\r\n" );
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document.write( "The second integer is N+1.  So let's multiply 3 times N+1. \r\n" );
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document.write( "[We have to put parentheses around (N+1)]\r\n" );
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document.write( "So to multiply that by 3 we have\r\n" );
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document.write( "3(N+1)\r\n" );
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document.write( "Now we use the distributive property to remove the parentheses:\r\n" );
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document.write( "3N + 3\r\n" );
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document.write( "Now we ask:  Did we get 3N + 3 both times?  The answer is yes.\r\n" );
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document.write( "Therefore we have proved that Bob is correct regardless of what N is.\r\n" );
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document.write( "Edwin
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