document.write( "Question 77779: What are the next five terms in the sequence:
\n" ); document.write( "2, 1/2, 2/9, 2/16, 2/25.....? \n" ); document.write( "
Algebra.Com's Answer #55780 by Edwin McCravy(20056) $\"\"$ $\"About$
You can put this solution on YOUR website!
\r\n" );
document.write( "What are the next five terms in the sequence:\r\n" );
document.write( "2, 1/2, 2/9, 2/16, 2/25.....?\r\n" );
document.write( "\r\n" );
document.write( "There are two methods to try.  Successive\r\n" );
document.write( "differences and successive ratios (quotients).\r\n" );
document.write( "One method is usually easier than the other.\r\n" );
document.write( "\r\n" );
document.write( "First I'll try doing it by successive differences \r\n" );
document.write( "to see if there is an easily recognizable \r\n" );
document.write( "pattern:\r\n" );
document.write( "\r\n" );
document.write( "2nd term - 1st term = 1/2 - 2 = 1/2 - 4/2 = -3/2\r\n" );
document.write( "\r\n" );
document.write( "3rd term - 2nd term = 2/9 - 1/2 = 4/18 - 9/18 = -5/18\r\n" );
document.write( "\r\n" );
document.write( "4th term - 3rd term = 2/16 - 2/9 = 1/8 - 2/9 = -7/72 \r\n" );
document.write( "\r\n" );
document.write( "5th term - 4th term = 2/25 - 2/16 = 2/25 - 1/8 = \r\n" );
document.write( "                            16/200 - 25/200 = -9/200\r\n" );
document.write( "\r\n" );
document.write( "The numerators are successive negative odd numbers,\r\n" );
document.write( "but the denominators do not form a very easily \r\n" );
document.write( "recognizable pattern. Maybe we could find a recognizable \r\n" );
document.write( "pattern for them, but it's not immediately obvious.  \r\n" );
document.write( "So let's turn to the other method, the successive ratios\r\n" );
document.write( "(quotients) to see if it is any easier.  If not we will \r\n" );
document.write( "have to come back to this one and try to find a \r\n" );
document.write( "recognizable pattern for those denominators.\r\n" );
document.write( "\r\n" );
document.write( "So let's check the successive ratios\r\n" );
document.write( "(quotients) to see if there is a more \r\n" );
document.write( "easily recognizable pattern:\r\n" );
document.write( "\r\n" );
document.write( "2nd term ÷ 1st term = 1/2 ÷ 2 =  1/2 × 1/2 = 1/4\r\n" );
document.write( "\r\n" );
document.write( "3rd term ÷ 2nd term = 2/9 ÷ 1/2 =  2/9 × 2/1 = 4/9\r\n" );
document.write( "\r\n" );
document.write( "4th term ÷ 3rd term = 2/16 ÷ 2/9 =  2/16 × 9/2 = 9/16\r\n" );
document.write( "\r\n" );
document.write( "5th term ÷ 4th term = 2/25 ÷ 2/16 =  2/25 × 16/2 = 16/25\r\n" );
document.write( "\r\n" );
document.write( "Yes we can easily recognize that pattern, because the\r\n" );
document.write( "numerators and denominators are the successive perfect\r\n" );
document.write( "squares.\r\n" );
document.write( "\r\n" );
document.write( "1/4 = 1²/2²\r\n" );
document.write( "\r\n" );
document.write( "4/9 = 2²/3²\r\n" );
document.write( "\r\n" );
document.write( "9/16 = 3²/4²\r\n" );
document.write( "\r\n" );
document.write( "16/25 = 4²/5²\r\n" );
document.write( "\r\n" );
document.write( "so we can extend the successive ratios\r\n" );
document.write( "\r\n" );
document.write( "1²/2², 2²/3², 3²/4², 4²/5²\r\n" );
document.write( "\r\n" );
document.write( "this way:\r\n" );
document.write( "\r\n" );
document.write( "1²/2², 2²/3², 3²/4², 4²/5², 5²/6², 6²/7², 7²/8², 8²/9², 9²/10²\r\n" );
document.write( "\r\n" );
document.write( "These are\r\n" );
document.write( "\r\n" );
document.write( "1/4, 4/9, 9/16, 16/25, 25/36, 36/49, 49/64, 64/81, 81/100\r\n" );
document.write( "\r\n" );
document.write( "So the sequence\r\n" );
document.write( "\r\n" );
document.write( "2, 1/2, 2/9, 2/16, 2/25\r\n" );
document.write( "\r\n" );
document.write( "can be extended to the next five terms this way:\r\n" );
document.write( "\r\n" );
document.write( "6th term = 5th term × 25/36 = 2/25 × 25/36 = 1/18\r\n" );
document.write( "\r\n" );
document.write( "7th term = 6th term × 36/49 = 1/18 × 36/49 = 2/49\r\n" );
document.write( "\r\n" );
document.write( "8th term = 7th term × 49/64 = 2/49 × 49/64 = 1/32\r\n" );
document.write( "\r\n" );
document.write( "9th term = 8th term × 64/81 = 1/32 × 64/81 = 2/81\r\n" );
document.write( "\r\n" );
document.write( "10th term = 9th term × 81/100 = 2/81 × 81/100 = 1/50\r\n" );
document.write( "\r\n" );
document.write( "So the answer is:\r\n" );
document.write( "\r\n" );
document.write( "2, 1/2, 2/9, 2/16, 2/25, 1/18, 2/49, 1/32, 2/81, 1/50\r\n" );
document.write( "\r\n" );
document.write( "In doing such problems, we should try both methods,\r\n" );
document.write( "to see which has the more easily recognizable pattern.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( " \n" ); document.write( "
\n" );