document.write( "Question 1082960: the 3rd and 4th terms of geometric progression are 12 and 8. find the sum to infinity progression. \n" ); document.write( "

Algebra.Com's Answer #696981 by KMST(5328) $\"\"$ $\"About$

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A geometric progression (called a geometric sequence in the USA) is a sequence of numbers (called terms) where the ratio of consecutive terms is always the same. That ratio is called the common ratio, $\"r\"$ .
\n" ); document.write( "In this case $\"r=8%2F12=2%2F3\"$ .
\n" ); document.write( "If we call the first term of our sequence $\"b\"$ ,
\n" ); document.write( "the first $\"n\"$ terms of the progression are
\n" ); document.write( " $\"b\"$ , $\"b%2Ar\"$ , $\"b%2Ar%5E2\"$ , $\"b%2Ar%5E3\"$ , ..., $\"b%0D%0A%2Ar%5E%28n-1%29\"$ .
\n" ); document.write( "The sum of those $\"n\"$ terms is
\n" ); document.write( "

.
\n" ); document.write( "When $\"r%3C1\"$ , $\"r-1%3C0%7D%7D+and+%7B%7B%7B%5E-1%3C0\"$ , so we like to write it as
\n" ); document.write( " $\"b%2A%28%281-r%5En%29%2F%281-r%29%29\"$ instead.
\n" ); document.write( "In that case, as $\"n\"$ increases, $\"r%5En\"$ becomes smaller,
\n" ); document.write( "tending to zero.
\n" ); document.write( "So the sum to infinity is $\"b%2A%28%281-0%29%2F%281-r%29%29=b%2F%281-r%29\"$ .
\n" ); document.write( "In this case, the third term is
\n" ); document.write( " $\"b%2Ar%5E2=12\"$ , or $\"b%2A%282%2F3%29%5E2=12\"$ , or $\"b%2A%284%2F9%29=12\"$ .
\n" ); document.write( "So, $\"b=12%2A9%2F4=27\"$ , and the sum to infinity is
\n" ); document.write( " $\"b%2F%281-r%29\"$ = $\"27%2F%28%281-2%2F3%29%29\"$ = $\"27%2F%28%281%2F3%29%29\"$ = $\"27%2A3=highlight%2881%29\"$ . \n" ); document.write( "

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