document.write( "Question 779653: a^(1/x)=b^(1/y)=c^(1/z) if a,b,c are in GP. Prove x,y,z are in AP \n" ); document.write( "
Algebra.Com's Answer #475309 by jsmallt9(3758)\"\" \"About 
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\n" ); document.write( "\"a%5E%281%2Fx%29=b%5E%281%2Fy%29=c%5E%281%2Fz%29\"
\n" ); document.write( "If a, b and c are in a geometric progression (which I assume \"GP\" means) then consecutive terms have a common (fixed) ratio. If we call this ratio \"r\" then
\n" ); document.write( "\"b%2Fa+=+r\"
\n" ); document.write( "or b = a*r
\n" ); document.write( "and
\n" ); document.write( "\"c%2Fb+=+r\"
\n" ); document.write( "or c = b*r

\n" ); document.write( "Since b = a*r, we can write c in terms of a:
\n" ); document.write( "\"c+=+%28a%2Ar%29%2Ar+=+a%2Ar%5E2\"

\n" ); document.write( "Substituting these expressions in \"a\" for \"b\" and \"c\" into the given equation we get:
\n" ); document.write( "\"a%5E%281%2Fx%29=%28a%2Ar%29%5E%281%2Fy%29=%28a%2Ar%5E2%29%5E%281%2Fz%29\"

\n" ); document.write( "To see if x, y and z are in an arithmetic progression (AP), where consecutive terms have a common (fixed) difference, we will start by expressing y and z in terms of x. First we'll do y:
\n" ); document.write( "\"a%5E%281%2Fx%29=%28a%2Ar%29%5E%281%2Fy%29\"
\n" ); document.write( "First let's eliminate the fractions in the exponents. Raising both sides to the LCD power:
\n" ); document.write( "\"%28a%5E%281%2Fx%29%29%5E%28x%2Ay%29=%28%28a%2Ar%29%5E%281%2Fy%29%29%5E%28x%2Ay%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"a%5Ey=%28a%2Ar%29%5Ex\"
\n" ); document.write( "Now we use logarithms to get the x's and y's out of the exponents. Finding the base a log of each side:
\n" ); document.write( "\"log%28a%2C+%28a%5Ey%29%29=log%28a%2C+%28%28a%2Ar%29%5Ex%29%29\"
\n" ); document.write( "Using a property of logs, the exponents in the arguments can be moved out in front:
\n" ); document.write( "\"y%2Alog%28a%2C+%28a%29%29=x%2Alog%28a%2C+%28a%2Ar%29%29\"
\n" ); document.write( "The log on the left is just a 1. On the right we can use another property of logs to split it into two logs (separating the \"a\" and the \"r\"):
\n" ); document.write( "\"y+=+x%2A%28log%28a%2C+%28a%29%29+%2B+log%28a%2C+%28r%29%29%29\"
\n" ); document.write( "The first log on the right is a 1 so this simplifies to:
\n" ); document.write( "\"y+=+x%281+%2B+log%28a%2C+%28r%29%29%29\"
\n" ); document.write( "\"y+=+x+%2B+x%2Alog%28a%2C+%28r%29%29\"

\n" ); document.write( "Now we repeat the process for z:
\n" ); document.write( "\"a%5E%281%2Fx%29=%28a%2Ar%5E2%29%5E%281%2Fz%29\"
\n" ); document.write( "\"%28a%5E%281%2Fx%29%29%5E%28x%2Az%29=%28%28a%2Ar%5E2%29%5E%281%2Fz%29%29%5E%28x%2Az%29\"
\n" ); document.write( "\"a%5Ez=%28a%2Ar%5E2%29%5Ex\"
\n" ); document.write( "\"log%28a%2C+%28a%5Ez%29%29=log%28a%2C+%28%28a%2Ar%5E2%29%5Ex%29%29\"
\n" ); document.write( "\"z%2Alog%28a%2C+%28a%29%29=x%2Alog%28a%2C+%28a%2Ar%5E2%29%29\"
\n" ); document.write( "\"z=x%2A%28log%28a%2C+%28a%29%29+%2B+log%28a%2C+%28r%5E2%29%29%29\"
\n" ); document.write( "\"z=x%2A%281+%2B+2%2Alog%28a%2C+%28r%29%29%29\"
\n" ); document.write( "\"z=x+%2B+2x%2Alog%28a%2C+%28r%29%29\"

\n" ); document.write( "If x, y and z are in an AP then they should have a common difference (which we will call \"d\"). Let's see:
\n" ); document.write( "\"d%5B1%5D+=+y+-+x\"
\n" ); document.write( "Substituting in for y:
\n" ); document.write( "\"d%5B1%5D+=+%28x+%2B+x%2Alog%28a%2C+%28r%29%29%29+-+x\"
\n" ); document.write( "The x's cancel:
\n" ); document.write( "\"d%5B1%5D+=+x%2Alog%28a%2C+%28r%29%29%29\"
\n" ); document.write( "Now lets try
\n" ); document.write( "\"d%5B2%5D+=+z+-+y\"
\n" ); document.write( "Substituting for both z and y:
\n" ); document.write( "\"d%5B2%5D+=+%28x+%2B+2x%2Alog%28a%2C+%28r%29%29%29+-+%28x+%2B+x%2Alog%28a%2C+%28r%29%29%29\"
\n" ); document.write( "Again the x's cancel:
\n" ); document.write( "\"d%5B2%5D+=+2x%2Alog%28a%2C+%28r%29%29+-+x%2Alog%28a%2C+%28r%29%29\"
\n" ); document.write( "These are like terms so we can subtract them:
\n" ); document.write( "\"d%5B2%5D+=+x%2Alog%28a%2C+%28r%29%29\"

\n" ); document.write( "As we can see, the two differences are the same. So x, y and z are in an AP. \n" ); document.write( "
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