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The sum of the first two terms of a geometric sequence is 90. The sum of the sixth and seventh terms is -10/27. Find the sum of the first seven terms of the sequence.
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\n" ); document.write( "I could not get the problem to work if the sum of the 6th and 7th terms is -10/27. If I assume the sum is actually 10/27, then the problem can be solved.
\n" ); document.write( "For a geometric sequence, the nth term can be written as:
\n" ); document.write( "a_n = ar^(n-1) where a = the 1st term, r = the common ratio
\n" ); document.write( "Then the sum of the 1st two terms can be written
\n" ); document.write( "90 = a + ar = a(1+r) [1]
\n" ); document.write( "The sum of the 6th and 7th terms can be written
\n" ); document.write( "10/27 = ar^5 + ar^6 = ar^5(1+r) [2]
\n" ); document.write( "From [1], 1+r = 90/a
\n" ); document.write( "Substitute this value into [2]:
\n" ); document.write( "10/27 = ar^5(90/a) = 90r^5
\n" ); document.write( "Solve for r:
\n" ); document.write( "r^5 = 10/(27*90)
\n" ); document.write( "r = (10(27*90))^(1/5)
\n" ); document.write( "This gives r = 1/3
\n" ); document.write( "Now we can solve for a:
\n" ); document.write( "90 = a(1+1/3) = a(4/3) -> a = 67.5
\n" ); document.write( "The sum of the first n terms of a geometric series is:
\n" ); document.write( "S_n = a(1-r^n)/(1-r)
\n" ); document.write( "So the sum of the first 7 terms is
\n" ); document.write( "So S_7 = 67.5(1-(1/3)^7)/(1-1/3) = 101.204 \n" ); document.write( "