document.write( "Question 723103: Three numbers form an arithmetic sequence having a common difference of 4, if the first number is increased by 2, the second number increased by 3 and the third number by 5, the resulting numbers form a geometric sequence. Find the original numbers. \n" ); document.write( "

Algebra.Com's Answer #443188 by KMST(5328) $\"\"$ $\"About$

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Let the number in arithmetic sequence with a common difference of 4 be
\n" ); document.write( " $\"x-4\"$ , $\"x\"$ , and $\"x%2B4\"$
\n" ); document.write( "I assume that by \"the first number\" the problem means the smallest of them, which would be $\"x-4\"$
\n" ); document.write( "If $\"x-4\"$ is increased by 2, we get
\n" ); document.write( " $\"x-4%2B2=x-2\"$ .
\n" ); document.write( "If $\"x\"$ (the second number) is increased by 3 , we get
\n" ); document.write( " $\"x%2B3\"$ .
\n" ); document.write( "If $\"x%2B4\"$ (the third number) is increased by 5, we get
\n" ); document.write( " $\"x%2B4%2B5=x%2B9\"$
\n" ); document.write( "Since the numbers $\"x-2\"$ , $\"x%2B3\"$ , and $\"x%2B9\"$ form a geometric sequence, the ratio of one number to the next is the same, meaning that
\n" ); document.write( " $\"%28x%2B9%29%2F%28x%2B3%29=%28x%2B3%29%2F%28x-2%29\"$
\n" ); document.write( "Equating the cross products (or, if you prefer, multiplying both sides of the equal sign times $\"%28x-2%29%28x%2B3%29\"$ to eliminate denominators) we get
\n" ); document.write( " $\"%28x%2B9%29%28x-2%29=%28x%2B3%29%5E2\"$ --> $\"x%5E2%2B7x-18=x%5E2%2B6x%2B9\"$ --> $\"7x-18=6x%2B9\"$ --> $\"7x-6x=9%2B18\"$ --> $\"highlight%28x=27%29\"$
\n" ); document.write( "The original numbers are:
\n" ); document.write( " $\"x-4=27-4=highlight%2823%29\"$
\n" ); document.write( " $\"x=highlight%2827%29\"$ and
\n" ); document.write( " $\"x%2B4=27%2B4=highlight%2831%29\"$ \n" ); document.write( "

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