document.write( "Question 825537: Hi, I have to find the general formula for the sequence 2 3 5 8 14 19 and so on, i.e. first 3-2=1, then 5-3=2, 8-5=3, etc, so the difference is simply 1,2,3,4,etc. That means the next number after 19 would be [(19-14)+1]+19 = 25, but the general formula a_n = ???\r
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Algebra.Com's Answer #497376 by AnlytcPhil(1806) $\"\"$ $\"About$

You can put this solution on YOUR website!
In a message dated 1/2/2014 10:50:51 A.M. Eastern Standard Time, devnull@algebra.com writes:
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\n" ); document.write( "Oh sorry, I messed up the calculations, of course it should be:
\n" ); document.write( "2 3 5 8 12 17 23 30 38 47 etc
\n" ); document.write( "so +1,+2,+3,+4,+5,+6,+7,+8,+9,etc
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document.write( "Hi,\r\n" );
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document.write( "The general term is\r\n" );
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document.write( "Here's what you do.  You list\r\n" );
document.write( "the numbers in a column like this:\r\n" );
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document.write( " 2\r\n" );
document.write( " 3\r\n" );
document.write( " 5\r\n" );
document.write( "12\r\n" );
document.write( "17\r\n" );
document.write( "23\r\n" );
document.write( "30\r\n" );
document.write( "38\r\n" );
document.write( "47\r\n" );
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document.write( "Then you make a column out beside that one, putting beside each number the\r\n" );
document.write( "difference between it and the number just below it, like this:\r\n" );
document.write( " \r\n" );
document.write( " 2  1\r\n" );
document.write( " 3  2\r\n" );
document.write( " 5  3\r\n" );
document.write( " 8  4\r\n" );
document.write( "12  5\r\n" );
document.write( "17  6\r\n" );
document.write( "23  7\r\n" );
document.write( "30  8\r\n" );
document.write( "38  9\r\n" );
document.write( "47\r\n" );
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document.write( "Then out beside that, you make a column of differences of differences:\r\n" );
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document.write( " 2  1  1\r\n" );
document.write( " 3  2  1\r\n" );
document.write( " 5  3  1\r\n" );
document.write( " 8  4  1\r\n" );
document.write( "12  5  1\r\n" );
document.write( "17  6  1\r\n" );
document.write( "23  7  1\r\n" );
document.write( "30  8  1\r\n" );
document.write( "38  9\r\n" );
document.write( "47\r\n" );
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document.write( "So it only takes 2 columns to get the differences to all be the same. So that\r\n" );
document.write( "means you have to assume a 2nd degree polynomial general term of the form:\r\n" );
document.write( "  \r\n" );
document.write( "y = ax²+bx+c\r\n" );
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document.write( "The unknowns will be a,b, and c. So you'll need to substitute three points\r\n" );
document.write( "(x,y) = (n,an), for n=1,2,3\r\n" );
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document.write( "(1,2), (2,3), (3,5)\r\n" );
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document.write( "2 = a(1)²+b(1)+c\r\n" );
document.write( "3 = a(2)²+b(2)+c\r\n" );
document.write( "5 = a(3)²+b(3)+c\r\n" );
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document.write( "That becomes:\r\n" );
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document.write( " a +  b + c = 2\r\n" );
document.write( "4a + 2b + c = 3\r\n" );
document.write( "9a + 3b + c = 5\r\n" );
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document.write( "Subtract the 1st eq. from the 2nd\r\n" );
document.write( "to eliminate c\r\n" );
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document.write( "3a + b = 1\r\n" );
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document.write( "Subtract the 2nd eq. from the 3rd\r\n" );
document.write( "to eliminate c.\r\n" );
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document.write( "5a + b = 2\r\n" );
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document.write( "Then subtract those equations to\r\n" );
document.write( "eliminate b\r\n" );
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document.write( "2a = 1\r\n" );
document.write( " a = 1/2\r\n" );
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document.write( "Then substitute in 3a + b = 1\r\n" );
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document.write( "3(1/2) + b = 1\r\n" );
document.write( "   3/2 + b = 1\r\n" );
document.write( "         b = -1/2\r\n" );
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document.write( "Then substitute in\r\n" );
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document.write( " a +  b + c = 2\r\n" );
document.write( "(1/2) + (-1/2) + c = 2\r\n" );
document.write( "          c = 2\r\n" );
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document.write( "So y = ax²+bx+c becomes\r\n" );
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document.write( "y = (1/2)x² - (1/2)x + 2\r\n" );
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document.write( "and you can write that as\r\n" );
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document.write( "y = (x²-x+4)/2\r\n" );
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document.write( "and change the x to n and the y to an\r\n" );
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document.write( "an = (n²-n+4)/2\r\n" );
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document.write( "You can do that with ANY sequence of numbers, but very often it takes more than\r\n" );
document.write( "just 2 columns to get them all the same.  For instance there is a 5th degree\r\n" );
document.write( "equation that would give you the general term for the sequence you made a\r\n" );
document.write( "mistake on.\r\n" );
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document.write( " 2  1  1  0  2  -8 \r\n" );
document.write( " 3  2  1  2 -6\r\n" );
document.write( " 5  3  3 -4\r\n" );
document.write( " 8  6 -1\r\n" );
document.write( "14  5\r\n" );
document.write( "19 \r\n" );
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document.write( "That takes 5 columns to get to a column\r\n" );
document.write( "that has \"all the same\" number. Yes that last column only has 1 number in it,\r\n" );
document.write( "-8, but \"all 'ONE' of it is 'the same'\". Hahaha :)\r\n" );
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document.write( "So there is a 5th degree polynomial\r\n" );
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document.write( "y=ax^5+bx^4+cx³+dx²+ex+f\r\n" );
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document.write( "There are 6 unknowns, so you'd have to substitute the six points\r\n" );
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document.write( "(1,2), (2,3), (3,5), (4,8), (5,14), (6,19) \r\n" );
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document.write( "in that and you'd have 6 equations in 6 unknowns.  Then you can eliminate f by\r\n" );
document.write( "subtracting, and get 5 equations in 5 unknowns and then get 4 equations in 4\r\n" );
document.write( "unknowns, then 3 in 3, and 2 in 2 and finally 1 equation in 1 unknown.  Then\r\n" );
document.write( "substitute back and you will get a very complicated 5th degree general term for\r\n" );
document.write( "the sequence you posted.\r\n" );
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document.write( "That will always work with any sequence.  But the general term will be\r\n" );
document.write( "complicated if there is no easy pattern.\r\n" );
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document.write( "Happy new year,\r\n" );
document.write( "Edwin

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