document.write( "Question 1100391: The sum of an infinite gp is 16 and the sum of the squares of its terms is 768/5. Find common ratio and 4th terms of progression. \n" ); document.write( "

Algebra.Com's Answer #714873 by ikleyn(52788) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "The sum of an infinite GP with the first term \"a\" and the common difference \"r\", |r| < 1 is  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, our first equation for the given GP is\r\n" );
document.write( "\r\n" );
document.write( " = 16.        (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The second GP, comprised of the squares of the first GP, has the first term  and the common difference .\r\n" );
document.write( "\r\n" );
document.write( "Therefore, our second equation for the sum of squares is\r\n" );
document.write( "\r\n" );
document.write( " = .      (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now divide eq(2) by eq(1) (both sides). You will get\r\n" );
document.write( "\r\n" );
document.write( " = .        (3)      (Take into account that  = (1-r)*(1+r))\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next step divide eq(3) by eq(1)  (both sides).  You will get\r\n" );
document.write( "\r\n" );
document.write( " = .         (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we are at the finish line.  From eq(4), making cross-multiplying, you get\r\n" );
document.write( "\r\n" );
document.write( "5*(1-r) = 3*(1+r)  ====>  5 - 5r = 3 + 3r  ====>  5-3 = 3r + 5r  ====>  8r = 2  ====>  r = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus we just found the common difference of the original progression.  It is  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Last step is to find \"a\":  a = 16*(1-r) =  = 8.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Answer.  The common ratio is ,  the first term is 8,  and the 4-th term is   = 1.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------
\n" ); document.write( "There is a bunch of lessons on geometric progressions in this site\r
\n" ); document.write( "\n" ); document.write( "    - Geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - The proofs of the formulas for geometric progressions \r
\n" ); document.write( "\n" ); document.write( "    - Problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Word problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - One characteristic property of geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Fresh, sweet and crispy problem on arithmetic and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction for sequences other than arithmetic or geometric\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Geometric progressions\".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "================
\n" ); document.write( "Thank you, Edwin.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );