document.write( "Question 732891: The first tem of a geometric progression is 7, its last term is 448 and its sum is 889. Find the common ratio \n" ); document.write( "

Algebra.Com's Answer #448267 by KMST(5328) $\"\"$ $\"About$

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THE LUCKY GUESS APPROACH:
\n" ); document.write( "If we call the number of terms $\"n\"$ , and the common ratio $\"r\"$ , the last term is
\n" ); document.write( " $\"b%5Bn%5D=7%2Ar%5E%28n-1%29=448\"$ --> $\"r%5E%28n-1%29=448%2F7\"$ --> $\"r%5E%28n-1%29=64\"$ -->
\n" ); document.write( " $\"r%5E%28n-1%29=2%5E6\"$
\n" ); document.write( "If we are lucky $\"r=2\"$ and $\"n-1=6\"$ --> $\"n=7\"$ .
\n" ); document.write( "Those numbers would certainly agree with the first term being 7 and the last term being 448.
\n" ); document.write( "Would the sum of those 7 terms be 889?
\n" ); document.write( "We can figure out the sum using some formula from the book, or we can calculate terms number 2, 3, 4, 5, and 6 and then add all 7 terms.
\n" ); document.write( "Either way, we would find out that the sum of the first 7 terms of a geometric progression with ratio 2 and first term 7 is indeed 889.
\n" ); document.write( "The approach worked because the problem was design to have nice small numbers as answers for $\"r\"$ and $\"n\"$ .
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\n" ); document.write( "ANOTHER APPROACH:
\n" ); document.write( "The sum of $\"n\"$ terms, from $\"b%5B1%5D=7\"$ to $\"b%5Bn%5D=448\"$ is $\"889\"$
\n" ); document.write( "The sum of the first $\"n-1\"$ terms, from $\"b%5B1%5D=7\"$ to $\"b%5Bn-1%5D\"$ is
\n" ); document.write( " $\"S=b%5B1%5D%2Bb%5B2%5D\"$ + ... + $\"b%5Bn-2%5D%2Bb%5Bn-1%5D=889-448=441\"$
\n" ); document.write( "If we call the common ratio $\"r\"$ ,
\n" ); document.write( " $\"S%2Ar=b%5B1%5D%2Ar%2Bb%5B2%5D%2Ar\"$ +... + $\"b%5Bn-2%5D%2Ar%2Bb%5Bn-1%5D%2Ar=b%5B2%5D%2Bb%5B3%5D\"$ + ... + $\"b%5Bn-1%5D%2Bb%5Bn%5D\"$
\n" ); document.write( "because $\"b%5B1%5D%2Ar=b%5B2%5D\"$ , $\"b%5B2%5D%2Ar=b%5B3%5D\"$ , and so on until $\"b%5Bn-2%5D%2Ar=b%5Bn-1%5D\"$ , and $\"b%5Bn-2%5D%2Ar=b%5Bn-1%5D\"$
\n" ); document.write( " $\"S%2Ar-S=b%5Bn%5D-b%5B1%5D\"$ --> $\"S%2A%28r-1%29=b%5Bn%5D-b%5B1%5D\"$ --> $\"r-1=%28b%5Bn%5D-b%5B1%5D%29%2FS\"$ --> $\"r=1%2B%28b%5Bn%5D-b%5B1%5D%29%2FS\"$
\n" ); document.write( "Since we know $\"b%5Bn%5D\"$ , $\"b%5B1%5D\"$ , and $\"S\"$ we can calculate
\n" ); document.write( " $\"r=1%2B%28448-7%29%2F441\"$ --> $\"r=1%2B441%2F441\"$ --> $\"r=1%2B1\"$ --> $\"highlight%28r=2%29\"$ \r
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