document.write( "Question 81178This question is from textbook
\n" ); document.write( ": find a formula for tn for the following:\r
\n" ); document.write( "\n" ); document.write( "0, -5, -10, -15...\r
\n" ); document.write( "\n" ); document.write( "1, -3, 9, -27...\r
\n" ); document.write( "\n" ); document.write( "2, 5, 10 ,17... \n" ); document.write( "
Algebra.Com's Answer #58194 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "find a formula for tn for the following: \r\n" );
document.write( "\r\n" );
document.write( "0, -5, -10, -15... \r\n" );
document.write( "\r\n" );
document.write( "Test to see whether it is an arithmetic or geometric sequence\r\n" );
document.write( "or neither:\r\n" );
document.write( "\r\n" );
document.write( "If it is arithmetic, then t2-t1 = t3-t2 = t4-t3 = d\r\n" );
document.write( "If it is geometric, then t2/t1 = t3/t2 = t4/t3 = r\r\n" );
document.write( "\r\n" );
document.write( "It is not geometric since we can't divide -5 by 0.\r\n" );
document.write( "\r\n" );
document.write( "Test for arithmetic sequence:\r\n" );
document.write( "\r\n" );
document.write( "t2-t1 = (-5)-(0) = -5\r\n" );
document.write( "t3-t2 = (-10)-(-5) = -10+5 = -5\r\n" );
document.write( "t4-t3 = (-15)-(-1) = -15+10 = -5\r\n" );
document.write( "\r\n" );
document.write( "These are all equal so it is an arithmetic sequence with\r\n" );
document.write( "common difference d = -5\r\n" );
document.write( "\r\n" );
document.write( "The formula for tn is\r\n" );
document.write( "\r\n" );
document.write( "tn = t1 + (n-1)d\r\n" );
document.write( "\r\n" );
document.write( "tn = 0 + (n-1)(-5)\r\n" );
document.write( "\r\n" );
document.write( "tn = -5(n-1)\r\n" );
document.write( "\r\n" );
document.write( "tn = -5n + 5\r\n" );
document.write( "\r\n" );
document.write( "tn = 5 - 5n \r\n" );
document.write( "\r\n" );
document.write( "--------------------------------\r\n" );
document.write( "\r\n" );
document.write( "1, -3, 9, -27... \r\n" );
document.write( "\r\n" );
document.write( "Test to see whether it is an arithmetic or geometric sequence\r\n" );
document.write( "or neither:\r\n" );
document.write( "\r\n" );
document.write( "If it is arithmetic, then t2-t1 = t3-t2 = t4-t3 = d\r\n" );
document.write( "If it is geometric, then t2/t1 = t3/t2 = t4/t3 = r\r\n" );
document.write( "\r\n" );
document.write( "Test for arithmetic sequence:\r\n" );
document.write( "\r\n" );
document.write( "t2-t1 = (-3)-(1) = -4\r\n" );
document.write( "t3-t2 = (9)-(-3) = 9+3 = 12\r\n" );
document.write( "t4-t3 = (-27)-(9) = -36\r\n" );
document.write( "\r\n" );
document.write( "These are not equal so it is not an arithmetic sequence.\r\n" );
document.write( "\r\n" );
document.write( "Test for geometric sequence:\r\n" );
document.write( "\r\n" );
document.write( "t2/t1 = (-3)/(1) = -3\r\n" );
document.write( "t3/t2 = (9)/(-3) = -3\r\n" );
document.write( "t4/t3 = (-27)/(9) = -3\r\n" );
document.write( "\r\n" );
document.write( "These are all equal so it is a geometric sequence with\r\n" );
document.write( "common ration r = -3\r\n" );
document.write( "\r\n" );
document.write( "The formula for tn is\r\n" );
document.write( "\r\n" );
document.write( "tn = t1rn-1\r\n" );
document.write( "\r\n" );
document.write( "tn = 1(-3)n-1\r\n" );
document.write( "\r\n" );
document.write( "tn = (-3)n-1\r\n" );
document.write( "\r\n" );
document.write( "------------------------------\r\n" );
document.write( "\r\n" );
document.write( "2, 5, 10 ,17...\r\n" );
document.write( "\r\n" );
document.write( "Test for arithmetic sequence:\r\n" );
document.write( "\r\n" );
document.write( "t2-t1 = (5)-(2) = 3\r\n" );
document.write( "t3-t2 = (10)-(5) = 5\r\n" );
document.write( "t4-t3 = (17)-(10) = 7\r\n" );
document.write( "\r\n" );
document.write( "These are not equal so it is not an arithmetic sequence.\r\n" );
document.write( "\r\n" );
document.write( "Test for geometric sequence:\r\n" );
document.write( "\r\n" );
document.write( "t2/t1 = (5)/(2) = 3/2\r\n" );
document.write( "t3/t2 = (10)/(5) = 2/3\r\n" );
document.write( "t4/t3 = (17)/(10) = 17/10\r\n" );
document.write( "\r\n" );
document.write( "These are not equal so it is not an geometric sequence either.\r\n" );
document.write( "\r\n" );
document.write( "When it is neither, we have to use our own intuition to see\r\n" );
document.write( "if we can relate it to some well-known sequence.\r\n" );
document.write( "\r\n" );
document.write( "If you look at the numbers carefully,\r\n" );
document.write( "\r\n" );
document.write( "2, 5, 10 ,17...\r\n" );
document.write( "\r\n" );
document.write( "and think about the common sequence of squares of integers:\r\n" );
document.write( "\r\n" );
document.write( "1, 4, 9, 16,...\r\n" );
document.write( "\r\n" );
document.write( "which has formula tn = n2\r\n" );
document.write( "\r\n" );
document.write( "you see that every number in 2, 5, 10 ,17... is just 1 more\r\n" );
document.write( "than the corresponding term in the sequence of squares of\r\n" );
document.write( "integers, so since the formula for tne sequence of squares \r\n" );
document.write( "is tn = n2, then the formula for the sequence every term of\r\n" );
document.write( "which is 1 more than the corresponding term of the sequence of\r\n" );
document.write( "squares of integers is\r\n" );
document.write( "\r\n" );
document.write( "tn = n2 + 1\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );