document.write( "Question 1071364: The first,third and sixth terms of an AP correspond to the first three consecutive terms of an increasing G.P.The first term of each progression is 16,the common difference is d and the common ratio of the G.P is r.
\n" ); document.write( "(I)Write two equations involving d and r
\n" ); document.write( "(ii)find the value of d and r
\n" ); document.write( "Find the sum of the first 20 terms
\n" ); document.write( "(I)the arithmetic progression
\n" ); document.write( "(ii)the geometric progression \n" ); document.write( "
Algebra.Com's Answer #686333 by KMST(5328)\"\" \"About 
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\"a%5Bn%5D=16%2B%28n-1%29%2Ad\" is term number \"n\" of the AP
\n" ); document.write( "\"b%5Bn%5D=16%2Ar%5E%28n-1%29\" is term number \"n\" of the GP
\n" ); document.write( "
\n" ); document.write( "(I) \"The first, third and sixth terms of an AP correspond to the first three consecutive terms of an increasing G.P.\" translates into the equationa
\n" ); document.write( "\"highlight%2816r=16%2B2d%29\" (1)
\n" ); document.write( "and
\n" ); document.write( "\"highlight%2816r%5E2=16%2B5d%29\" (2).
\n" ); document.write( "(ii) Adding equation (1) times \"-5\" plus equation (2) times \"2\" we get
\n" ); document.write( "\"32r%5E2-80r=-48\"
\n" ); document.write( "Rearranging,
\n" ); document.write( "\"32r%5E2-80r%2B48=0\"
\n" ); document.write( "Dividing both sides of the equal sign by \"16\"
\n" ); document.write( "\"2r%5E2-5r%2B3=0\"
\n" ); document.write( "Factoring, we get
\n" ); document.write( "\"%28r-1%29%282r-3%29=0\" ,
\n" ); document.write( "so the solutions are \"r=1\" and \"r=3%2F2\" .
\n" ); document.write( "\"r=1\" would make all terms of the GP equal to is the only reasonable answer.
\n" ); document.write( "Substituting into equation (1), we get
\n" ); document.write( "\"16%283%2F2%29=16%2B2d\"
\n" ); document.write( "\"24=16%2B2d\"
\n" ); document.write( "\"24-16=2d\"
\n" ); document.write( "\"8=2d\"
\n" ); document.write( "\"8%2F2=d\"
\n" ); document.write( "\"highlight%28d=4%29\"
\n" ); document.write( "
\n" ); document.write( "Find the sum of the first 20 terms:
\n" ); document.write( "(I) The sum of the first \"n\" terms of an AP can be calculated as
\n" ); document.write( "\"S%5Bn%5D=%282a%5B1%5D%2B%28n-1%29d%29n%2F2\" , where \"a%5B1%5D\" is the first term.
\n" ); document.write( "With \"system%28a%5B1%5D=16%2Cd=4%2Cn=20%29\" ,
\n" ); document.write( "\"S%5B20%5D=%282%2A16%2B%2820-1%294%2920%2F2\"
\n" ); document.write( "\"S%5B20%5D=%2832%2B19%2A4%2910\"
\n" ); document.write( "\"S%5B20%5D=%2832%2B76%2910\"
\n" ); document.write( "\"S%5B20%5D=108%2A10\"
\n" ); document.write( "\"highlight%28S%5B20%5D=1080%29\"
\n" ); document.write( "(ii) The sum of the first \"n\" terms of a GP can be calculated as
\n" ); document.write( "\"S%5Bn%5D=b%5B1%5D%28r%5En-1%29%2F%28r-1%29\" , where \"b%5B1%5D\" is the first term.
\n" ); document.write( "With \"system%28b%5B1%5D=16%2Cr=3%2F2%2Cn=20%29\" ,
\n" ); document.write( "\"S%5B20%5D=16%28%283%2F2%29%5E20-1%29%2F%283%2F2-1%29\"
\n" ); document.write( "\"S%5B20%5D=16%28%283%2F2%29%5E20-1%29%2F%281%2F2%29\"
\n" ); document.write( "\"S%5B20%5D=16%28%283%2F2%29%5E20-1%29%2A2\"
\n" ); document.write( "\"S%5B20%5D=2%5E4%28%283%2F2%29%5E20-1%29%2A2\"
\n" ); document.write( "\"S%5B20%5D=2%5E5%283%5E20%2F2%5E20-1%29\"
\n" ); document.write( "\"S%5B20%5D=2%5E5%28%283%5E20-2%5E20%29%2F2%5E20%29\"
\n" ); document.write( "\"S%5B20%5D=%283%5E20-2%5E20%29%2F2%5E15\"
\n" ); document.write( "We could ask a calculator and get
\n" ); document.write( "\"highlight%28S%5B20%5D=3485735825%2F32768%29\"
\n" ); document.write( "There is no way to simplify that fraction, and
\n" ); document.write( "as a decimal there would be 15 digits after the decimal point,
\n" ); document.write( "but an approximate value ois \"106376.2\" .
\n" ); document.write( "Just for fun, if you like factoring,
\n" ); document.write( "it is also \"S%5B20%5D=5%5E2%2A11%2A13%2A211%2A4621%2F2%5E15\" \n" ); document.write( "
\n" );