document.write( "Question 1116337: the fourth term of a GP is 1/2 and the sixth term is 1/8. its common ratio is negative
\n" ); document.write( "a) how many term of GP add up to give a sum of -2/22/32?
\n" ); document.write( "b)what is the sum to infinity for this GP \n" ); document.write( "

Algebra.Com's Answer #731215 by KMST(5328) $\"\"$ $\"About$

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ONE WAY TO START:
\n" ); document.write( "The recursive relation for a geometric progression,
\n" ); document.write( "relating term number $\"n\"$ , $\"a%5Bn%5D\"$ to the preceding term, $\"a%5Bn-1%5D\"$ is
\n" ); document.write( " $\"a%5Bn%5D=a%5Bn-1%5D%2Ar\"$ , where $\"r\"$ is the common ratio.
\n" ); document.write( "So, applying that to the fourth, fifth and sixth terms
\n" ); document.write( " $\"a%5B5%5D=a%5B4%5D%2Ar=%281%2F2%29%2Ar\"$ ,
\n" ); document.write( " $\"a%5B6%5D=a%5B5%5D%2Ar=%28%281%2F2%29%2Ar%29%2Ar=%281%2F2%29r%5E2\"$ , and
\n" ); document.write( " $\"%281%2F2%29r%5E2=1%2F8\"$ --> $\"r%5E2=%281%2F8%29%2F%281%2F2%29\"$ --> $\"r%5E2=1%2F4\"$ .
\n" ); document.write( "Knowing that $\"r%3C0\"$ , we conclude that $\"r=-1%2F2\"$ ,
\n" ); document.write( "and can write the first 4 terms, starting from the 4th,
\n" ); document.write( "and dividing by $\"%28-1%2F2%29\"$ (multiplying times -2)
\n" ); document.write( "each time to get the term before:
\n" ); document.write( " $\"a%5B4%5D=1%2F2\"$ ,
\n" ); document.write( " $\"a%5B3%5D=%281%2F2%29%2F%28-1%2F2%29=1%2F2%29%28-2%29=-1\"$ ,
\n" ); document.write( " $\"a%5B2%5D=%28-1%29%28-2%29=2\"$ ,
\n" ); document.write( " $\"a%5B1%5D=1%28-2%29=-4\"$ .
\n" ); document.write( "
\n" ); document.write( "ANOTHER WAY:
\n" ); document.write( "The general \"formula\" for term number $\"n\"$ , $\"a%5Bn%5D\"$ ,
\n" ); document.write( "of a GP with first term $\"a%5B1%5D\"$ and common ratio $\"r\"$ is
\n" ); document.write( " $\"a%5Bn%5D=a%5B1%5D%2Ar%5E%28n-1%29\"$ .
\n" ); document.write( "Applying that to $\"system%28a%5B4%5D=1%2F2%2Ca%5B6%5D=1%2F8%2Cr%3C0%29\"$ ,
\n" ); document.write( " $\"system%28a%5B1%5D%2Ar%5E3=1%2F2%2Ca%5B4%5D%2Ar%5E6=1%2F8%2Cr%3C0%29\"$ --> $\"system%28a%5B1%5D%2Ar%5E3=1%2F2%2Cr%5E2=%281%2F8%29%2F%281%2F2%29=1%2F4%2Cr%3C0%29\"$ --> $\"system%28a%5B1%5D%2A%28-1%2F2%29%5E3=1%2F2%2Cr=-1%2F2%29\"$ --> $\"system%28a%5B1%5D%2A%28-1%2F8%29=1%2F2%2Cr=-1%2F2%29\"$ --> $\"system%28a%5B1%5D=-4%2Cr=-1%2F2%29\"$
\n" ); document.write( "
\n" ); document.write( "AND FROM THERE:
\n" ); document.write( "The \"formula\" for the sum of the first $\"n\"$ terms, $\"S%5Bn%5D\"$ , of a GP is
\n" ); document.write( " $\"S%5Bn%5D=a%5B1%5D%28r%5En-1%29%2F%28r-1%29\"$
\n" ); document.write( "When $\"r%3C1\"$ ,
\n" ); document.write( "the term $\"r%5En\"$ tends to $\"0\"$ as $\"n\"$ tends to $\"infinity\"$ ,
\n" ); document.write( "and the sum to infinity is
\n" ); document.write( " $\"S%5Binfinity%5D=%28-a%5B1%5D%29%2F%281-r%29\"$ .
\n" ); document.write( "In this case the sum to infinity is easy to calculate:
\n" ); document.write( " $\"S%5Binfinity%5D\"$ $\"%22=%22\"$ $\"%28-%28-4%29%29%2F%28%28-1%2F2-1%29%29\"$ $\"%22=%22\"$ $\"4%2F%28%28-3%2F2%29%29\"$ $\"%22=%22\"$ $\"4%28-2%2F3%29\"$ $\"%22=%22\"$ $\"highlight%28-8%2F3=-2%262%2F3=%22-2.6666...%22%29\"$
\n" ); document.write( "Substituting the $\"a%5B1%5D\"$ and $\"r\"$ values found above,
\n" ); document.write( "the expression for the sum of the first $\"n\"$ terms becomes
\n" ); document.write( " $\"S%5Bn%5D\"$ $\"%22=%22\"$ $\"%28-%28-4%29%29%28%28-1%2F2%29%5En-1%29%2F%28-3%2F2%29\"$ $\"%22=%22\"$ $\"4%28-2%2F3%29%28%28-1%2F2%29%5En-1%29\"$ $\"%22=%22\"$ $\"%288%2F3%29%28%28%28-1%29%5En-2%5En%29%2F2%5En%29\"$ $\"%22=%22\"$ $\"%282%5E3%2F3%29%28%28%28-1%29%5En-2%5En%29%2F2%5En%29\"$ $\"%22=%22\"$ $\"%281%2F2%5E%28n-3%29%29%28%28%28-1%29%5En-2%5En%29%2F3%29\"$
\n" ); document.write( "If that is supposed to be an irreducible fraction of the form $\"K%2F32\"$ ,
\n" ); document.write( "the only possibility is $\"2%5E%28n-3%29=32\"$ ,
\n" ); document.write( "and as $\"32=2%5E5\"$ , $\"n-3=5\"$ --> $\"highlight%28n=8%29\"$ .
\n" ); document.write( "
\n" ); document.write( "BOTH QUESTIONS WERE ANSWERED, BUT IF YOU WANT MORE,
\n" ); document.write( "knowing $\"n\"$ , we could try to find that denominator.
\n" ); document.write( "

.
\n" ); document.write( "So,
\n" ); document.write( " $\"S%5Binfinity%5D\"$ $\"%22=%22\"$

. \n" ); document.write( "

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