document.write( "Question 383150: Let z1, z2 and z3 be three complex numbers in geometric progression. Suppose that the
\n" ); document.write( "average of these numbers is 10, while the average of their squares is 20i. Determine the
\n" ); document.write( "value of z2, the middle term. \n" ); document.write( "

Algebra.Com's Answer #271385 by robertb(5830) $\"\"$ $\"About$

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let z1 = a
\n" ); document.write( "z2 = ar, and
\n" ); document.write( " $\"z3+=+ar%5E2\"$ . Then from the given,
\n" ); document.write( " $\"%28a%2Bar%2Bar%5E2%29%2F3+=+10\"$ , and $\"%28a%5E2%2Ba%5E2r%5E2+%2B+a%5E2r%5E4%29%2F3+=+20i\"$ , both arising from the given.
\n" ); document.write( "Hence
\n" ); document.write( " $\"a%2Bar%2Bar%5E2+=+30\"$ , (1) and
\n" ); document.write( " $\"a%5E2%281%2Br%5E2+%2B+r%5E4%29+=+60i\"$ . (2)
\n" ); document.write( "Divide (2) by (1), to get
\n" ); document.write( " $\"%28a%5E2%281+%2B+r%5E2+%2B+r%5E4%29%29%2F%28a%281%2Br%2Br%5E2%29%29+=+%2860i%29%2F%2830%29\"$ , or
\n" ); document.write( " $\"a%281-r+%2B+r%5E2%29+=+2i\"$ , or $\"a-ar+%2B+ar%5E2+=+2i\"$ . Subtract this result from
\n" ); document.write( " $\"a%2Bar%2Bar%5E2+=+30\"$ , to get
\n" ); document.write( " $\"2ar+=+30+-+2i\"$ , OR, $\"ar+=+15+-+i\"$ . BUT ar is exactly the middle term of the geometric sequence of 3 geometric terms, or z2.
\n" ); document.write( "Thus, z2 = 15 - i. \n" ); document.write( "

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