document.write( "Question 978756: Find a sequence of four numbers. The first of which is six and the fourth of which is 16, if the first three numbers form an arithmetic sequence and the last three numbers form a geomtrice sequence \n" ); document.write( "

Algebra.Com's Answer #600169 by htmentor(1343) $\"\"$ $\"About$

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The sequence of numbers is 6, b, c, 16
\n" ); document.write( "The first three numbers form an arithmetic sequence:
\n" ); document.write( "b = 6 + d and c = 6 + 2d, where d = the common difference
\n" ); document.write( "The last three numbers form a geometric sequence. The first number of the sequence is b:
\n" ); document.write( "c = br and 16 = br^2, where r = the common ratio
\n" ); document.write( "r = c/b = (6+2d)/(6+d)
\n" ); document.write( "16 = br^2 = (6+d)((6+2d)/(6+d))^2 = (6+2d)^2/(6+d)
\n" ); document.write( "Solve for d:
\n" ); document.write( "(36+24d+4d^2)/(6+d) = 16
\n" ); document.write( "36 + 24d + 4d^2 = 96 + 16d
\n" ); document.write( "4d^2 + 8d - 60 = 0
\n" ); document.write( "d^2 + 2d - 15 = 0
\n" ); document.write( "(d-3)(d+5) = 0
\n" ); document.write( "This gives two possible answers, d=3, d=-5
\n" ); document.write( "So the two sequences are 6, 9, 12, 16 or 6, 1, -4, 16
\n" ); document.write( "The first sequence has common ratio c/b = 12/9 = 4/3
\n" ); document.write( "The second sequence has common ratio c/b = -4/1 = -4
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