document.write( "Question 942185: Q.3 Find a particular solution to the recurrence.
\n" ); document.write( "an+1 − 2an + an−1 = 5 + 2n, n ³ 1(15 marks) plz send full solution \r
\n" ); document.write( "\n" ); document.write( "Q.5 (a) If R = {(1, 1), (2, 1), (3, 2), (4, 3)}, find R2,R4.
\n" ); document.write( "(b) How many permutations are there of the letters, taken all at a
\n" ); document.write( "time, of the word ALLAHABAD?(7,7marks) plz send full solution\r
\n" ); document.write( "\n" ); document.write( "Q.6 Let A = {0, 1, 2, 3} and R = ((x, y) : x − y = 3k, k is an integer) i.e,
\n" ); document.write( "XRy if f x-y is divisible by 3, then prove that R is an equivalence
\n" ); document.write( "relation \n" ); document.write( "
Algebra.Com's Answer #575822 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Q.3 Find a particular solution to the recurrence.
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\r\n" );
document.write( "Let \"a%5B0%5D=a%5B1%5D=0\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5Bn%2B1%5D+-+2a%5Bn%5D+%2B+a%5Bn-1%5D+=+5+%2B+2n\"\r\n" );
document.write( "\r\n" );
document.write( "Let n=1\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B2%5D+-+2a%5B1%5D+%2B+a%5B0%5D+=+5+%2B+2%281%29\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B2%5D+-+2%280%29+%2B+0+=+7\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B2%5D+=+7\"\r\n" );
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document.write( "Let n=2\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B3%5D+-+2a%5B2%5D+%2B+a%5B1%5D+=+5+%2B+2%282%29\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B3%5D+-+2%287%29+%2B+0+=+9\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B3%5D+-+14+%2B+0+=+9\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B3%5D+=+23\"\r\n" );
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document.write( "Let n=3\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B4%5D+-+2a%5B3%5D+%2B+a%5B2%5D+=+5+%2B+2%283%29\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B4%5D+-+2%2823%29+%2B+7+=+11\", substituting (1),\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B4%5D+-+46+%2B+7+=+11\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B4%5D+-+39+=+11\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B4%5D+=+50\"\r\n" );
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document.write( "Let n=4\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B5%5D+-+2a%5B4%5D+%2B+a%5B3%5D+=+5+%2B+2%284%29\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B5%5D+-+2%2850%29+%2B+23+=+13\", substituting (1),\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B5%5D+-+46+%2B+7+=+11\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B5%5D+-+39+=+11\"\r\n" );
document.write( "\r\n" );
document.write( "     \"a%5B5%5D+=+50\"\r\n" );
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document.write( "Now we make a difference table\r\n" );
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document.write( " 0   0   7   2\r\n" );
document.write( " 0   7   9   2\r\n" );
document.write( " 7  16  11   2\r\n" );
document.write( "23  27  13 \r\n" );
document.write( "50  40\r\n" );
document.write( "90\r\n" );
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document.write( "It required the 3rd difference to get all\r\n" );
document.write( "constants so we assume a 3rd degree polynomial\r\n" );
document.write( "for this particular solution:\r\n" );
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document.write( "\"a%5Bn%5D=An%5E3%2BBn%5E2%2BCn%2BD\"\r\n" );
document.write( "\"a%5B0%5D=A%280%29%5E3%2BB%280%29%5E2%2BC%280%29%2BD\"\r\n" );
document.write( "\"0=D\"\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=An%5E3%2BBn%5E2%2BCn\"\r\n" );
document.write( "\"a%5B1%5D=A%281%29%5E3%2BB%281%29%5E2%2BC%281%29\"\r\n" );
document.write( "\"0=A%2BB%2BC\"\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=An%5E3%2BBn%5E2%2BCn\"\r\n" );
document.write( "\"a%5B2%5D=A%282%29%5E3%2BB%282%29%5E2%2BC%282%29\"\r\n" );
document.write( "\"7=8A%2B4B%2B2C\"\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=An%5E3%2BBn%5E2%2BCn\"\r\n" );
document.write( "\"a%5B3%5D=A%283%29%5E3%2BB%283%29%5E2%2BC%283%29\"\r\n" );
document.write( "\"23=27A%2B9B%2B3C\"\r\n" );
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document.write( "Thuse we have the system\r\n" );
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document.write( "\"system%28A%2BB%2BC=0%2C8A%2B4B%2B2C=7%2C27A%2B9B%2B3C=23%29\"\r\n" );
document.write( " \r\n" );
document.write( "That has solution \"A=1%2F3\", \"B=5%2F2\", \"C=-17%2F6\",\r\n" );
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document.write( "and we found \"D=0\" earlier.\r\n" );
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document.write( "so we suspect that a particular solution is\r\n" );
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document.write( "\"a%5Bn%5D=expr%281%2F3%29n%5E3%2Bexpr%285%2F2%29n%5E2-expr%2817%2F6%29n%2B0\"\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=%282n%5E3%2B15n%5E2-17n%29%2F6\"\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=%28n%282n%5E2%2B15n-17%29%29%2F6\"\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=%28n%28n-1%29%282n%2B17%29%29%2F6\"\r\n" );
document.write( "\r\n" );
document.write( "--------------------------------\r\n" );
document.write( "

\n" ); document.write( "R = {(1,1), (2,1), (3,2), (4,3)}, find R2,R4
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document.write( "R2 is set of all ordered pair of ordered pairs in R: =\r\n" );
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document.write( "I'll use brackets to indicate an ordered pair of ordered pairs:\r\n" );
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document.write( "{[(1,1),(1,1)], [(1,1),(2,1)], [(1,1),(3,2)], [(1,1),(4,3)], \r\n" );
document.write( " [(2,1),(1,1)], [(2,1),(2,1)], [(2,1),(3,2)], [(2,1),(4,3)], \r\n" );
document.write( " [(3,2),(1,1)], [(3,2),(2,1)], [(3,1),(3,2)], [(3,1),(4,3)], \r\n" );
document.write( " [(4,3),(1,1)], [(4,3),(2,1)], [(4,1),(3,2)], [(4,1),(4,3)]}\r\n" );
document.write( "

\n" ); document.write( "R4
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document.write( "No way Jose!  That's the set of every ordered pair of those ordered \r\n" );
document.write( "pairs of ordered pairs. It's ridiculous of your teacher to expect \r\n" );
document.write( "you to list 256 ordered pairs of ordered pairs of ordered pairs of\r\n" );
document.write( "ordered pairs.  Any teacher that would ask a student to do that \r\n" );
document.write( "should be fired!  They're not qualified to teach! That's not \r\n" );
document.write( "teaching. That's student abuse!\r\n" );
document.write( "

\n" ); document.write( "(b) How many permutations are there of the letters, taken all at a
\n" ); document.write( "time, of the word ALLAHABAD?
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document.write( "ALLAHABAD has 9 letters.  If you could tell the A's apart and the L's apart,\r\n" );
document.write( "then the answer would be 9!, but since you can't tell them apart,\r\n" );
document.write( "you must: \r\n" );
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document.write( "1. divide by 4! for that's how many ways the A's can be permuted\r\n" );
document.write( "in any one permutation, and they would all look just alike.  \r\n" );
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document.write( "and in addition to that, you will also have to\r\n" );
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document.write( "2. divide by 2! for that's how many ways the L's can be permuted\r\n" );
document.write( "in any one permutation, and they would both look just alike.  \r\n" );
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document.write( "Answer: \"9%21%2F%284%212%21%29=7560\" ways.\r\n" );
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document.write( "That's all I'm going to do now.\r\n" );
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document.write( "Edwin
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