document.write( "Question 922288: A sequence of numbers is said to form a harmonic progression provided their reciprocals form an arithmetic progression. Insert three harmonic means between - 1/2 and 1/14. \n" ); document.write( "

Algebra.Com's Answer #559471 by KMST(5328) $\"\"$ $\"About$

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I understand the first term to be the positive number $\"1%2F2\"$ .
\n" ); document.write( "We need an arithmetic progression, where each term differs from the one before by a common difference $\"d\"$ .
\n" ); document.write( "The first term is $\"2\"$ , which is followed by 3 more terms, and then by $\"14\"$ .
\n" ); document.write( "The terms are $\"2\"$ , $\"2%2Bd\"$ , $\"2%2B2d\"$ , $\"2%2B3d\"$ , $\"14\"$
\n" ); document.write( "The fifth term is $\"2%2B%285-1%29d=2%2B4d=14\"$ .
\n" ); document.write( " $\"2%2B4d=14\"$ ---> $\"4d=14-2\"$ ---> $\"4d=12\"$ ---> $\"d=12%2F4\"$ ---> $\"d=3\"$ .
\n" ); document.write( "So the terms of the arithmetic progression are
\n" ); document.write( " $\"2\"$ , $\"2%2B3=5\"$ , $\"5%2B3=8\"$ , $\"8%2B3=11\"$ , $\"11%2B3=14\"$ ,
\n" ); document.write( "and the terms of the harmonic progression are
\n" ); document.write( " $\"1%2F2\"$ , $\"1%2F5\"$ , $\"1%2F8\"$ , $\"1%2F11\"$ , and $\"1%2F14\"$ .
\n" ); document.write( "
\n" ); document.write( "NOTE: If the first term was $\"-1%2F2\"$ ,
\n" ); document.write( "then the arithmetic progression would be
\n" ); document.write( " $\"-2\"$ , $\"2\"$ , $\"6\"$ , $\"10\"$ , $\"14\"$ ,
\n" ); document.write( "and the harmonic progression would be
\n" ); document.write( " $\"-1%2F2\"$ , $\"1%2F2\"$ , $\"1%2F6\"$ , $\"1%2F10\"$ , $\"1%2F14\"$ . \n" ); document.write( "

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