document.write( "Question 1073135: Given that $\"1%2F%28y-x%29\"$ , $\"1%2F%282y%29\"$ and $\"1%2F%28y-z%29\"$ are consecutive terms of an arithmetric progression, prove that x, y and z are consecutive terms of a geometric progression \n" ); document.write( "

Algebra.Com's Answer #688031 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
One nice thing about arithmetic progressions is that
\n" ); document.write( "if a, b, and c are three consecutive terms in an arithmetic progression (AP),
\n" ); document.write( "a + c = 2b (and if a + c =2b, a, b, and c form an AP).
\n" ); document.write( "That means that
\n" ); document.write( " $\"1%2F%28y-x%29%2B1%2F%28y-z%29=2%281%2F2y%29\"$
\n" ); document.write( " $\"%28%28y-z%29%2B%28y-x%29%29%2F%28%28y-x%29%28y-z%29%29=1%2Fy\"$
\n" ); document.write( " $\"%282y-x-z%29%2F%28y%5E2-xy-zy%2Bxz%29=1%2Fy\"$
\n" ); document.write( " $\"y%282y-x-z%29=y%5E2-xy-zy%2Bxz\"$
\n" ); document.write( " $\"2y%5E2-xy-zy=y%5E2-xy-zy%2Bxz\"$
\n" ); document.write( "Cancelling like terms,
\n" ); document.write( " $\"2y%5E2=y%5E2%2Bxz\"$
\n" ); document.write( " $\"highlight%28y%5E2=xz%29\"$
\n" ); document.write( "
\n" ); document.write( "One nice thing about geometric progressions is that
\n" ); document.write( "if x, y, and z are three consecutive terms in a geometric progression (GP),
\n" ); document.write( " $\"xz=y%5E2\"$ , and if $\"xz=y%5E2\"$ then x, y, and z form a GP,
\n" ); document.write( "and that is exactly what we found above. \n" ); document.write( "

\n" );