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the ratio appears to be -2
\n" ); document.write( "1/24 * -2 = -1/12
\n" ); document.write( "-1/12 * -2 = 1/6
\n" ); document.write( "1/6 * -2 = -1/3, etc.
\n" ); document.write( "the formula for a geometric series with n terms is:
\n" ); document.write( "S[n] = a * (1-r^n)/(1-r)
\n" ); document.write( "a is the first term in the sequence.
\n" ); document.write( "r is the common ratio.
\n" ); document.write( "for example:
\n" ); document.write( "assume a is equal to 10 and r is equal to 5 and n is equal to 2.
\n" ); document.write( "s[2] = a * (1-r^2)/(1-r) = 10 * (1-5^2)/(1-5) = 60
\n" ); document.write( "it's easy to see that the sum for this sequence is equal to 60 because:
\n" ); document.write( "a = 10
\n" ); document.write( "a*r = 10*5 = 50
\n" ); document.write( "sum of a + a*r = 60
\n" ); document.write( "applying this to your problem, this formula should provide us with the correct answer.
\n" ); document.write( "in your problem:
\n" ); document.write( "a = 1/24
\n" ); document.write( "r = -2
\n" ); document.write( "n = 5
\n" ); document.write( "the formula becomes:
\n" ); document.write( "s[5] = 1/24 * (1-(-2)^5)/(1-(-2))
\n" ); document.write( "this becomes:
\n" ); document.write( "s[5] = 1/24 * (33)/(3) = 1/24 * 11 = 11/24
\n" ); document.write( "we can see if this is correct by simply adding up the terms.
\n" ); document.write( "a = 1/24 (first term)
\n" ); document.write( "1/24 * (-2) = -1/12 (second term)
\n" ); document.write( "-1/12 * (-2) = 1/6 (third term)
\n" ); document.write( "1/6 * (-2) = -1/3 (fourth term)
\n" ); document.write( "-1/3 * (-2) = 2/3 (fifth term)
\n" ); document.write( "add these terms up together and you get:
\n" ); document.write( "1/24 - 1/12 + 1/6 - 1/3 + 2/3
\n" ); document.write( "convert everything to a fraction with a denominator of 24 to get:
\n" ); document.write( "1/24 - 2/24 + 4/24 - 8/24 + 16/24
\n" ); document.write( "this becomes:
\n" ); document.write( "(1 - 2 + 4 - 8 + 16) / 24 which becomes:
\n" ); document.write( "(21 - 10) / 24 which becomes:
\n" ); document.write( "11/24.
\n" ); document.write( "the formula works.
\n" ); document.write( "the answer is 11/24.
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