document.write( "Question 988607: The fourth term of a geometric progression exceeds the third term by 54, and the sum of the second and third term is 36. Find the common ratio if it is positive \n" ); document.write( "

Algebra.Com's Answer #609119 by KMST(5328) $\"\"$ $\"About$

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$\"r\"$ = the common ratio.
\n" ); document.write( " $\"a%5B2%5D\"$ = the second term.
\n" ); document.write( " $\"a%5B3%5D=a%5B2%5D%2Ar\"$ = the third term.
\n" ); document.write( " $\"a%5B3%5D=a%5B3%5D%2Ar=a%5B2%5D%2Ar%5E2\"$ = the fourth term.
\n" ); document.write( "The fourth term of a geometric progression exceeds the third term by 54 translates as
\n" ); document.write( " $\"a%5B4%5D-a%5B3%5D=54\"$ <--> $\"a%5B2%5D%2Ar%5E2-a%5B2%5D%2Ar=54\"$ <--> $\"a%5B2%5D%2A%28r%5E2-r%29=54\"$ .
\n" ); document.write( "The sum of the second and third term is 36 translates as
\n" ); document.write( " $\"a%5B2%5D%2Ba%5B3%5D=36\"$ <--> $\"a%5B2%5D%2Ba%5B2%5D%2Ar=36\"$ <--> $\"a%5B2%5D%2A%281%2Br%29=36\"$ .
\n" ); document.write( "The ratio both equations tells us that
\n" ); document.write( " $\"%28a%5B2%5D%2A%28r%5E2-r%29%29%2F%28a%5B2%5D%2A%281%2Br%29%29=54%2F36\"$ --> $\"%28r%5E2-r%29%2F%281%2Br%29=3%2F2\"$ --> $\"2%28r%5E2-r%29=3%281%2Br%29\"$ --> $\"2r%5E2-2r%29=3%2B3r\"$ }--> $\"2r%5E2-5r-3=0\"$ .
\n" ); document.write( "The equation $\"2r%5E2-5r-3=0\"$ is a quadratic equation.
\n" ); document.write( "As is true for all quadratic equations, it can be solved by \"completing the square, or by using the quadratic formula.
\n" ); document.write( "This particular quadratic equation can also be solved by factoring:
\n" ); document.write( " $\"2r%5E2-5r-3=0\"$ ---> $\"2r%5E2-6r%2Br-3=0\"$ ---> $\"2r%28r-3%29%2Br-3=0\"$ }---> $\"%282r%2B1%29%28r-3%29=0\"$ ---> $\"system%28highlight%28r=3%29%2C%22or%22%2Cr=-1%2F2%29\"$ .
\n" ); document.write( "If the common ratio is positive, it must be $\"highlight%28r=3%29\"$ . \n" ); document.write( "

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