document.write( "Question 529087: Consider the sequence x-3, x+1, 2x+8. One value for x is 5, making the sequence geometric.
\n" ); document.write( "find the other value of x for which the sequence is geometric
\n" ); document.write( "For this value of x find the common ratio and the sum of the infinite sequence \n" ); document.write( "

Algebra.Com's Answer #349489 by KMST(5328) $\"\"$ $\"About$

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For the sequence to be geometric the ratio between consecutive terms must be the same.
\n" ); document.write( " $\"%28x%2B1%29%2F%28x-3%29=%282x%2B8%29%2F%28x%2B1%29\"$
\n" ); document.write( "You could use that definition to find your $\"x\"$ .
\n" ); document.write( "Or you could use the fact that in a geometric sequence each middle term is the geometic mean of the neighboring terms. If a, b, and c are consecutive terms in a geometric sequence that means
\n" ); document.write( " $\"b=sqrt%28ac%29\"$ or $\"b%5E2=ac\"$
\n" ); document.write( "Either way, you end up with
\n" ); document.write( " $\"%28x%2B1%29%5E2=%282x%2B8%29%28x-3%29\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"x%5E2=25\"$
\n" ); document.write( "so the solutions are
\n" ); document.write( " $\"x=5\"$ and $\"x=-5\"$
\n" ); document.write( "That makes the first three terms -8, -4, and -2
\n" ); document.write( "and you should get the ratio and sum from that easily.
\n" ); document.write( "Without even using the formula for sum of a geometric sequence I realize that
\n" ); document.write( "-8+(-4)+(-2)+(-1)+(-1/2)+ ... gets closer and closer to -16, and the difference is always equal to that shrinking last term. \n" ); document.write( "

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