document.write( "Question 1018511: if a1,a2,a3.... are in A.P Such tgat a2/a7=3/2, then the 13th term of A.P is
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #634650 by ikleyn(52786) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "If a1,a2,a3.... are in A.P such that a2/a7=3/2, then the 13th term of A.P is
\n" ); document.write( "------------------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Please check your condition.\r\n" );
document.write( "\r\n" );
document.write( "I was able to conclude that the 17th term is zero. \r\n" );
document.write( "\r\n" );
document.write( "But there is no info to guess the 13th term.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "OK, we have a disputed issue.
\n" ); document.write( "Let me put my arguments.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "1.\r\n" );
document.write( "\r\n" );
document.write( "We have  = ,   = ,\r\n" );
document.write( "where  is the first term of the AP and d is the common difference.\r\n" );
document.write( "\r\n" );
document.write( "We also have an equation \r\n" );
document.write( "\r\n" );
document.write( " = ,          (1)\r\n" );
document.write( "\r\n" );
document.write( "which is the same as \r\n" );
document.write( "\r\n" );
document.write( " = .    (2)\r\n" );
document.write( "\r\n" );
document.write( "From (2) we have\r\n" );
document.write( "\r\n" );
document.write( " = ,   or\r\n" );
document.write( "\r\n" );
document.write( " = ,    or\r\n" );
document.write( "\r\n" );
document.write( " = .    (3)\r\n" );
document.write( "\r\n" );
document.write( "The left side,  ,  is nothing else as  .\r\n" );
document.write( "\r\n" );
document.write( "Therefore, (3) means that  = ,\r\n" );
document.write( "exactly as I said at the very beginning of my post.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2.\r\n" );
document.write( "\r\n" );
document.write( "Now, my second statement is: from the given condition we can not make any conclusion about the concrete value of any term of this AP, \r\n" );
document.write( "    unless it is zero.\r\n" );
document.write( "\r\n" );
document.write( "Indeed, we can multiply all terms of the given AP by an arbitrary non-zero number. The condition (1) still will be held, but the \r\n" );
document.write( "    concrete value of , let say, will be multiplied by this arbitrary factor.\r\n" );
document.write( "\r\n" );
document.write( "Only if the term of the AP is zero, it will not be changed after multiplication. \r\n" );
document.write( "So, based on given conditions, we can make quantitative suggestion only about the zero term of the AP. \r\n" );
document.write( "

\n" ); document.write( " \n" ); document.write( "

\n" );