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find the first 3 terms of a geometric progression whose sum is 42 and whose product is 512
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\n" ); document.write( "The nth term of a geometric sequence is
\n" ); document.write( "a(n) = ar^(n-1) where a = the first term, r=the common ratio
\n" ); document.write( "The 1st 3 terms are a, ar, and ar^2
\n" ); document.write( "The sum of these terms is 42:
\n" ); document.write( "a + ar + ar^2 = 42 = a(1+r+r^2) [1]
\n" ); document.write( "The product is 512:
\n" ); document.write( "a*ar*ar^2 = 512 = (ar)^3 [2]
\n" ); document.write( "The last equation gives r = 8/a. Substitute this value in [1]:
\n" ); document.write( "a(1 + 8/a + 64/a^2) = 42
\n" ); document.write( "Simplifying gives a^2 - 34a + 64 = 0
\n" ); document.write( "Factor:
\n" ); document.write( "(a-32)(a-2) = 0
\n" ); document.write( "This gives a=2, a=32
\n" ); document.write( "Using the 1st solution, we have r=4
\n" ); document.write( "So one such progression satisfying the conditions is a(n) = 2(4)^(n-1),
\n" ); document.write( "which gives 2,8,32 as the 1st 3 terms
\n" ); document.write( "[If the other solution a=32 were used, the 1st 3 terms would be reversed(r=1/4): 32,8,2]
\n" ); document.write( "Either way, the sum and products are the same. \n" ); document.write( "