document.write( "Question 1113863: The first three consecutive terms of an exponential sequence are (x-1), 2x and (5x+3) respectively. 1.find the value of x. 2.find the common ratio. 3.find the sum of the first six terms. \n" ); document.write( "

Algebra.Com's Answer #728922 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "The ratio between the second and first terms is the same as the ratio between the third and second terms:

\n" ); document.write( " $\"2x%2F%28x-1%29+=+%285x%2B3%29%2F%282x%29\"$
\n" ); document.write( " $\"5x%5E2-2x-3+=+4x%5E2\"$
\n" ); document.write( " $\"x%5E2-2x-3+=+0\"$
\n" ); document.write( " $\"%28x-3%29%28x%2B1%29+=+0\"$

\n" ); document.write( " $\"x+=+3\"$ or $\"x+=+-1\"$

\n" ); document.write( "Both values of x produce geometric sequences; but one of them is not very interesting:

\n" ); document.write( "x=3: 2, 6, 18, ...
\n" ); document.write( "x=-1: -2, -2, -2, ...

\n" ); document.write( "For x=3, the common ratio is 3, and the sum of the first 6 terms is 2+6+18+54+162+486 = 728.

\n" ); document.write( "Since we only needed to find the sum of the first 6 terms, it was easy simply to find the terms and add them. We could have used the formula for the sum of a finite geometric sequence. Since the formula is useful when we need to find the sum of a large number of terms, it is a useful formula to know.

\n" ); document.write( " $\"S%28n%29+=+%28a%281-r%5En%29%29%2F%281-r%29\"$
\n" ); document.write( "where a is the first term and r is the common ratio.

\n" ); document.write( "For this problem,
\n" ); document.write( " $\"S%286%29+=+%282%281-3%5E6%29%29%2F%281-3%29+=+%282%28-728%29%29%2F%28-2%29+=+728\"$

\n" ); document.write( "The problem is not very interesting for the case where x=-1; in that case, the common ratio is 1, and the sum of the first 6 terms is 6(-2) = -12. \n" ); document.write( "

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