document.write( "Question 872658: a1 = 24^1/3, an+1=(an+24)^1/3, n ≥ 1. Then the integer part of a100 equals \r
\n" ); document.write( "\n" ); document.write( "A. 2
\n" ); document.write( "B. 10
\n" ); document.write( "C. 100
\n" ); document.write( "D. 24
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #526441 by KMST(5328) $\"\"$ $\"About$

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$\"8%3C24%3C27\"$ <---> $\"2=8%5E%28%221+%2F+3%22%29%3C24%5E%28%221+%2F+3%22%29%3C27%5E%28%221+%2F+3%22%29=3\"$
\n" ); document.write( "So $\"2%3Ca%5B1%5D%3C3\"$ meaning that $\"2%3Ca%5Bn%5D%3C3\"$ is true for $\"n=1\"$
\n" ); document.write( "Could it be true for all positive integer $\"n\"$ values.
\n" ); document.write( "If it were true for $\"n=h\"$ , we would know that $\"0%3Ca%5Bh%5D%3C3\"$ .
\n" ); document.write( "Then $\"24%3Ca%5Bh%5D%2B24%3C3%2B24\"$
\n" ); document.write( " $\"24%3C%28a%5Bh%5D%2B24%29%3C27\"$
\n" ); document.write( " $\"24%5E%281%2F3%29%3C%28a%5Bh%5D%2B24%29%5E%281%2F3%29%3C27%5E%281%2F3%29=3\"$ ,
\n" ); document.write( "and since $\"24%5E%28%221+%2F+3%22%29=a%5B1%5D\"$ and $\"%28a%5Bh%5D%2B24%29%5E%281%2F3%29=a%5Bh%2B1%5D\"$ ,
\n" ); document.write( "we conclude that $\"2%3Ca%5B1%5D%3Ca%5Bh%2B1%5D%3C3\"$ .
\n" ); document.write( "Since $\"2%3Ca%5Bn%5D%3C3\"$ is true for $\"n=1\"$ ,
\n" ); document.write( "and being true for $\"n=h\"$ makes it true for $\"n=h%2B1\"$ ,
\n" ); document.write( "then $\"2%3Ca%5Bn%5D%3C3\"$ is true for all values of $\"n\"$ .
\n" ); document.write( "The integer part of $\"a%5Bn%5D\"$ for any $\"n\"$ is $\"2\"$ ,
\n" ); document.write( "and so the integer part of $\"a%5B100%5D\"$ is $\"highlight%282%29\"$ .
\n" ); document.write( "It is true that $\"a%5Bn%5D\"$ gets ever closer to $\"3\"$ , but it can never get there. \n" ); document.write( "

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