Algebra.Com's Answer #774089 by ikleyn(52810)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( " It is a nice problem, but formulated INACCURATELY.\r
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document.write( " An accurate formulation is THIS :\r
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document.write( " There is a mountain with 30 bat caves in a row that contain 340 bats in all. \r\n" );
document.write( " Any 7 caves in a row contain exactly 77 bats. \r\n" );
document.write( " Suppose the first cave has 7 times  as many bats  as the last cave. \r\n" );
document.write( " How many bats are in the 29th cave?\r\n" );
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document.write( " Why it must be so, and why it can not be in different way, you will see, when you complete reading my solution.\r
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document.write( "Solution\r
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document.write( "Let me start my solution from the reference to an Internet dictionary\r
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document.write( " \"in a row\" means \"in succession\".\r
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document.write( "In this problem, it is the only meaning of this term.\r
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document.write( "1. Since any 7 caves in a row contain exactly 77 bats, we have, in particular\r\n" );
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document.write( "  = 77, (1)\r\n" );
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document.write( "  = 77. (2)\r\n" );
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document.write( " Subtract equation (1) from equation (2). The terms from  to  will cancel each other, and you will get\r\n" );
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document.write( "  -  = 0, or\r\n" );
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document.write( " = .\r\n" );
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document.write( " Making the same step by step, you will get\r\n" );
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document.write( " =  ,\r\n" );
document.write( "  =  ,\r\n" );
document.write( "  =  ,\r\n" );
document.write( "  =  ,\r\n" );
document.write( " =  ,\r\n" );
document.write( "  =  ,\r\n" );
document.write( " =  = \r\n" );
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document.write( " and so on. It means that the sequence , , , . . . ,  is PERIODICAL with the period 7.\r\n" );
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document.write( " It is the FIRST IDEA of the solution.\r\n" );
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document.write( "2. Its consequence is that =  .\r\n" );
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document.write( " Therefore, from the condition, we have this equation\r\n" );
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document.write( " = =  . (3)\r\n" );
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document.write( " It is the SECOND IDEA of the solution.\r\n" );
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document.write( "3. From the condition, we have \r\n" );
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document.write( " = 77,\r\n" );
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document.write( "  = 77,\r\n" );
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document.write( "  = 77,\r\n" );
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document.write( "  = 77.\r\n" );
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document.write( " since any 7 caves in a row contain exactly 77 bats.\r\n" );
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document.write( " Add these 4 equation (both sides). You will get\r\n" );
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document.write( " + + + . . . +  = 4*77 = 308.\r\n" );
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document.write( " It means that \r\n" );
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document.write( " + = 340 - 308 = 32. (4)\r\n" );
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document.write( " It is the THIRD IDEA of the solution.\r\n" );
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document.write( "4. Now, from equations (3) and (4) we have\r\n" );
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document.write( " + = 32.\r\n" );
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document.write( " It implies  = 32, or\r\n" );
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document.write( " = 32/8 = 4.\r\n" );
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document.write( " Hence, = = 7*4 = 28.\r\n" );
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document.write( "ANSWER. = 28. The 29-th cave has 28 bats.\r\n" );
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document.write( "Solved.\r
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document.write( "---------------\r
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document.write( "The solution by @MathLover1 is INCORRECT.\r
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document.write( "It is incorrect starting from its second line, where @Mathlover1 states that it follows from the condition
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document.write( "that each cave has 11 bats.\r
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document.write( "This statement is WRONG, and it DOES NOT FOLLOW from the condition.\r
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